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Біноміальний коефіцієнт — Вікіпедія
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коефіцієнт</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Матеріал з Вікіпедії — вільної енциклопедії.</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти до навігації</a> <a class="mw-jump-link" href="#searchInput">Перейти до пошуку</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="uk" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Pascal%27s_triangle_5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/200px-Pascal%27s_triangle_5.svg.png" decoding="async" width="200" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/300px-Pascal%27s_triangle_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/400px-Pascal%27s_triangle_5.svg.png 2x" data-file-width="540" data-file-height="389" /></a><figcaption>Біноміальні коефіцієнти розташовані у вигляді <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA_%D0%9F%D0%B0%D1%81%D0%BA%D0%B0%D0%BB%D1%8F" title="Трикутник Паскаля">трикутника Паскаля</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Binomial_theorem_visualisation.svg" class="mw-file-description"><img alt="Візуалізація біноміальних коефіцієнтів до 4 степеня" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/229px-Binomial_theorem_visualisation.svg.png" decoding="async" width="229" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/344px-Binomial_theorem_visualisation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/458px-Binomial_theorem_visualisation.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Візуалізація біноміальних коефіцієнтів до 4 степеня</figcaption></figure> <p><b>Біноміальні коефіцієнти</b> — коефіцієнти в розкладі <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (1+x)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (1+x)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788b53636e1f3c6c9ceaad92f26e5d4a689f7a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle \ (1+x)^{n}}"></span> по степенях <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9923068b5684340554217dfc8b2b968523527d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.91ex; height:1.676ex;" alt="{\displaystyle \ x}"></span> (так званий <a href="/wiki/%D0%91%D1%96%D0%BD%D0%BE%D0%BC_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0" title="Біном Ньютона">біном Ньютона</a>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{n}={n \choose 0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots +{n \choose n}x^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}={n \choose 0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots +{n \choose n}x^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d3bc58e276d6371f1b43938a33fefa79ba9e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.791ex; height:7.009ex;" alt="{\displaystyle (1+x)^{n}={n \choose 0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots +{n \choose n}x^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}.}"></span></dd></dl> <p>Біноміальний коефіцієнт є узагальненням кількості <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%86%D1%96%D1%8F_(%D0%BA%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0)" title="Комбінація (комбінаторика)">невпорядкованих виборів</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b088b67f00e4739e57c658ac7dd5913898013ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.843ex;" alt="{\displaystyle C_{n}^{k}}"></span>, що визначена тільки для невід'ємних <a href="/wiki/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" class="mw-redirect" title="Цілі числа">цілих чисел</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, тобто <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n+1\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n+1\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6826d89e41f6e58fc692184f4a501aecc1d97c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.497ex; height:2.343ex;" alt="{\displaystyle \ n+1\in \mathbb {N} }"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k+1\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k+1\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82dea1d2b29352e39840ca33d2a71701e7775cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.96ex; height:2.343ex;" alt="{\displaystyle \ k+1\in \mathbb {N} .}"></span> У явному вигляді для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo>≤<!-- ≤ --></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq k\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b429c3c44b3dc10332285272eee6f754dbf985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq k\leq n}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}=C_{n}^{k}={\frac {n!}{k!\;(n-k)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}=C_{n}^{k}={\frac {n!}{k!\;(n-k)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35854cec48e637330a5a57823725c9e0cd56245b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.141ex; height:6.343ex;" alt="{\displaystyle {n \choose k}=C_{n}^{k}={\frac {n!}{k!\;(n-k)!}}}"></span>,</dd></dl> <p>де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949e312c9bf9a9a5e641c1db1b1d7c6f0425b536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k!}"></span> — <a href="/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D1%96%D0%B0%D0%BB" title="Факторіал">факторіали</a> чисел <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> і <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="uk" dir="ltr"><h2 id="mw-toc-heading">Зміст</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Явні_формули"><span class="tocnumber">1</span> <span class="toctext">Явні формули</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Трикутник_Паскаля"><span class="tocnumber">2</span> <span class="toctext">Трикутник Паскаля</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Властивості"><span class="tocnumber">3</span> <span class="toctext">Властивості</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Твірні_функції"><span class="tocnumber">3.1</span> <span class="toctext">Твірні функції</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Подільність"><span class="tocnumber">3.2</span> <span class="toctext">Подільність</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Тотожності"><span class="tocnumber">3.3</span> <span class="toctext">Тотожності</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Біном_Ньютона_і_наслідки"><span class="tocnumber">3.4</span> <span class="toctext">Біном Ньютона і наслідки</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Згортка_Вандермонда_і_наслідки"><span class="tocnumber">3.5</span> <span class="toctext">Згортка Вандермонда і наслідки</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Інші_тотожності"><span class="tocnumber">3.6</span> <span class="toctext">Інші тотожності</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Матричні_співвідношення"><span class="tocnumber">3.7</span> <span class="toctext">Матричні співвідношення</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Цілозначні_многочлени"><span class="tocnumber">3.8</span> <span class="toctext">Цілозначні многочлени</span></a></li> <li class="toclevel-2 tocsection-12"><a href="#Асимптотика_і_оцінки"><span class="tocnumber">3.9</span> <span class="toctext">Асимптотика і оцінки</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-13"><a href="#Алгоритми_обчислення"><span class="tocnumber">4</span> <span class="toctext">Алгоритми обчислення</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Узагальнення"><span class="tocnumber">5</span> <span class="toctext">Узагальнення</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Генерація_на_C++"><span class="tocnumber">6</span> <span class="toctext">Генерація на C++</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Див._також"><span class="tocnumber">7</span> <span class="toctext">Див. також</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Примітки"><span class="tocnumber">8</span> <span class="toctext">Примітки</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#Джерела"><span class="tocnumber">9</span> <span class="toctext">Джерела</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Явні_формули"><span id=".D0.AF.D0.B2.D0.BD.D1.96_.D1.84.D0.BE.D1.80.D0.BC.D1.83.D0.BB.D0.B8"></span>Явні формули</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=1" title="Редагувати розділ: Явні формули" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=1" title="Редагувати вихідний код розділу: Явні формули"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Обчислюючи коефіцієнти в розкладі <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04845cc4a998213c15d61b36512d27222fca3527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle (1+x)^{n}}"></span> у степеневий ряд, можна отримати явні формули для біноміальних коефіцієнтів <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20897631d805059d3e86b791c9d6b96c0f20abf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle \textstyle {\binom {n}{k}}}"></span>. </p><p>Для всіх дійсних чисел <i>n</i> і цілих чисел <i>k</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n(n-1)(n-2)\cdot \ldots \cdot (n-k+1)}{k!}},&k\geqslant 0,\\0,&k<0,\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mo>…<!-- … --></mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>k</mi> <mo>⩾<!-- ⩾ --></mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>k</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n(n-1)(n-2)\cdot \ldots \cdot (n-k+1)}{k!}},&k\geqslant 0,\\0,&k<0,\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807264f4a6c0e8ac41fddf391d4eebad3b5eab38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.523ex; height:7.509ex;" alt="{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n(n-1)(n-2)\cdot \ldots \cdot (n-k+1)}{k!}},&k\geqslant 0,\\0,&k<0,\end{cases}}}"></span></dd></dl> <p>де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949e312c9bf9a9a5e641c1db1b1d7c6f0425b536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k!}"></span> позначає <a href="/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D1%96%D0%B0%D0%BB" title="Факторіал">факторіал</a> числа <i>k</i>. </p><p>Для невід'ємних цілих <i>n</i> і <i>k</i> також справедливі формули: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n!}{k!(n-k)!}},&0\leqslant k\leqslant n,\\0,&k>n.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mn>0</mn> <mo>⩽<!-- ⩽ --></mo> <mi>k</mi> <mo>⩽<!-- ⩽ --></mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>k</mi> <mo>></mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n!}{k!(n-k)!}},&0\leqslant k\leqslant n,\\0,&k>n.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1065ec9e22f5d085882f81e33caf810e3450411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.129ex; height:7.509ex;" alt="{\displaystyle {\binom {n}{k}}={\begin{cases}{\frac {n!}{k!(n-k)!}},&0\leqslant k\leqslant n,\\0,&k>n.\end{cases}}}"></span></dd></dl> <p>Для цілих від'ємних показників коефіцієнти розкладу бінома <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae77acbd60171666a3fe844970fe7bbe860b7a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.639ex; height:3.009ex;" alt="{\displaystyle (1+x)^{-n}}"></span> рівні </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {-n}{k}}={\begin{cases}(-1)^{k}\cdot {\frac {(n+k-1)!}{k!(n-1)!}},&k\geqslant 0,\\0,&k<0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−<!-- − --></mo> <mi>n</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>k</mi> <mo>⩾<!-- ⩾ --></mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>k</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {-n}{k}}={\begin{cases}(-1)^{k}\cdot {\frac {(n+k-1)!}{k!(n-1)!}},&k\geqslant 0,\\0,&k<0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f3b0255a160a51f79541d96c7f13f4edeae0bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.778ex; height:7.509ex;" alt="{\displaystyle {\binom {-n}{k}}={\begin{cases}(-1)^{k}\cdot {\frac {(n+k-1)!}{k!(n-1)!}},&k\geqslant 0,\\0,&k<0.\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Трикутник_Паскаля"><span id=".D0.A2.D1.80.D0.B8.D0.BA.D1.83.D1.82.D0.BD.D0.B8.D0.BA_.D0.9F.D0.B0.D1.81.D0.BA.D0.B0.D0.BB.D1.8F"></span>Трикутник Паскаля</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=2" title="Редагувати розділ: Трикутник Паскаля" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=2" title="Редагувати вихідний код розділу: Трикутник Паскаля"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" style="padding-left:20px"><i>Докладніше: <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA_%D0%9F%D0%B0%D1%81%D0%BA%D0%B0%D0%BB%D1%8F" title="Трикутник Паскаля">Трикутник Паскаля</a></i></div> <p>Тотожність <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203b128a098e18cbb8cf36d004bd7282b28461bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.392ex; height:6.176ex;" alt="{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}"></span> дозволяє розташувати біноміальні коефіцієнти для невід'ємних <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> у вигляді трикутника Паскаля, в якому кожне число рівне сумі двох, що стоять на рядок вище: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}n=0:&&&&&1&&&&\\n=1:&&&&1&&1&&&\\n=2:&&&1&&2&&1&&\\n=3:&&1&&3&&3&&1&\\n=4:&1&&4&&6&&4&&1\\\vdots &&\vdots &&\vdots &&\vdots &&\vdots &\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>:</mo> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> </mtr> <mtr> <mtd> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>:</mo> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd /> <mtd /> </mtr> <mtr> <mtd> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>:</mo> </mtd> <mtd /> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mn>2</mn> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>:</mo> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>:</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mn>4</mn> </mtd> <mtd /> <mtd> <mn>6</mn> </mtd> <mtd /> <mtd> <mn>4</mn> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd /> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd /> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd /> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd /> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}n=0:&&&&&1&&&&\\n=1:&&&&1&&1&&&\\n=2:&&&1&&2&&1&&\\n=3:&&1&&3&&3&&1&\\n=4:&1&&4&&6&&4&&1\\\vdots &&\vdots &&\vdots &&\vdots &&\vdots &\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0b06f720b51e85ee972be3934456eeebc961a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:39.065ex; height:20.176ex;" alt="{\displaystyle {\begin{matrix}n=0:&&&&&1&&&&\\n=1:&&&&1&&1&&&\\n=2:&&&1&&2&&1&&\\n=3:&&1&&3&&3&&1&\\n=4:&1&&4&&6&&4&&1\\\vdots &&\vdots &&\vdots &&\vdots &&\vdots &\end{matrix}}}"></span></dd></dl> <p>Трикутна таблиця, запропонована <a href="/wiki/%D0%9F%D0%B0%D1%81%D0%BA%D0%B0%D0%BB%D1%8C_%D0%91%D0%BB%D0%B5%D0%B7" class="mw-redirect" title="Паскаль Блез">Паскалем</a> в «Трактаті про арифметичний трикутник» (<a href="/wiki/1654" title="1654">1654</a>), відрізняється від описаної тут поворотом на 45°. Таблиці для зображення біноміальних коефіцієнтів були відомі й раніше. </p> <div class="mw-heading mw-heading2"><h2 id="Властивості"><span id=".D0.92.D0.BB.D0.B0.D1.81.D1.82.D0.B8.D0.B2.D0.BE.D1.81.D1.82.D1.96"></span>Властивості</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=3" title="Редагувати розділ: Властивості" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=3" title="Редагувати вихідний код розділу: Властивості"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Твірні_функції"><span id=".D0.A2.D0.B2.D1.96.D1.80.D0.BD.D1.96_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D1.96.D1.97"></span>Твірні функції</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=4" title="Редагувати розділ: Твірні функції" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=4" title="Редагувати вихідний код розділу: Твірні функції"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Для фіксованого значення <i>n</i> <a href="/wiki/%D0%93%D0%B5%D0%BD%D0%B5%D1%80%D0%B0%D1%82%D1%80%D0%B8%D1%81%D0%B0" title="Генератриса">твірною функцією послідовності</a> біноміальних коефіцієнтів <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{0}},\;{\tbinom {n}{1}},\;{\tbinom {n}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{0}},\;{\tbinom {n}{1}},\;{\tbinom {n}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62f4b8eaf9ab94bf55d548ce97a474f418e006aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.353ex; height:3.343ex;" alt="{\displaystyle {\tbinom {n}{0}},\;{\tbinom {n}{1}},\;{\tbinom {n}{2}},}"></span> … є </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\binom {n}{k}}x^{k}=(1+x)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\binom {n}{k}}x^{k}=(1+x)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03aa43d17f3612feab7254823ff51e6ca1ef9b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.082ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\binom {n}{k}}x^{k}=(1+x)^{n}.}"></span></dd></dl> <p>Для фіксованого значення <i>k</i> твірною функцією послідовності коефіцієнтів <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {0}{k}},\;{\tbinom {1}{k}},\;{\tbinom {2}{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>0</mn> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>1</mn> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>2</mn> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {0}{k}},\;{\tbinom {1}{k}},\;{\tbinom {2}{k}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422440a8aae929c1c71bccc0e0422a62340456c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.964ex; height:3.343ex;" alt="{\displaystyle {\tbinom {0}{k}},\;{\tbinom {1}{k}},\;{\tbinom {2}{k}},}"></span> … є </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n}{\binom {n}{k}}y^{n}={\frac {y^{k}}{(1-y)^{k+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n}{\binom {n}{k}}y^{n}={\frac {y^{k}}{(1-y)^{k+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ac24c7fb5dea9aa2499fe12c209fa210312fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.028ex; height:7.009ex;" alt="{\displaystyle \sum _{n}{\binom {n}{k}}y^{n}={\frac {y^{k}}{(1-y)^{k+1}}}}"></span>.</dd></dl> <p>Двовимірною твірною функцією біноміальних коефіцієнтів <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206415d3742167e319b2e52c2ca7563b799abad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}}"></span> для цілих <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfaf07760c32f6bda7900a7b8bbcad8eb49e6dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.64ex; height:2.509ex;" alt="{\displaystyle n,k}"></span> є </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>y</mi> <mo>−<!-- − --></mo> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15727bea9a18c1e59b8f45d45a1732495ff455a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.421ex; height:7.009ex;" alt="{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}},}"></span> або <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>y</mi> <mo>−<!-- − --></mo> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beed430d1886de453bb8817b8b0ca5f076809cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.163ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-y-xy}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Подільність"><span id=".D0.9F.D0.BE.D0.B4.D1.96.D0.BB.D1.8C.D0.BD.D1.96.D1.81.D1.82.D1.8C"></span>Подільність</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=5" title="Редагувати розділ: Подільність" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=5" title="Редагувати вихідний код розділу: Подільність"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Із <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9B%D1%8E%D0%BA%D0%B0" title="Теорема Люка">теореми Люка</a> випливає, що: </p> <ul><li>коефіцієнт <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20897631d805059d3e86b791c9d6b96c0f20abf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle \textstyle {\binom {n}{k}}}"></span> непарний <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></span> в <a href="/wiki/%D0%94%D0%B2%D1%96%D0%B9%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Двійкова система числення">двійкового запису</a> числа <i>k</i> одиниці не стоять у тих розрядах, де в числі <i>n</i> стоять нулі;</li> <li>коефіцієнт <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20897631d805059d3e86b791c9d6b96c0f20abf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle \textstyle {\binom {n}{k}}}"></span> <a href="/wiki/%D0%9F%D0%BE%D0%B4%D1%96%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Подільність">не кратний</a> <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Просте число">простому число</a> <i>p</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></span> при записі числа <i>k</i> в <a href="/wiki/%D0%9F%D0%BE%D0%B7%D0%B8%D1%86%D1%96%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Позиційна система числення">системі числення з основою</a> <i>p</i>, всі розряди не перевищують відповідних розрядів числа <i>n</i>;</li> <li>у послідовності біноміальних коефіцієнтів <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {n}{0}},\;{\binom {n}{1}},\;\ldots ,\;{\binom {n}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…<!-- … --></mo> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n}{0}},\;{\binom {n}{1}},\;\ldots ,\;{\binom {n}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5570d62264adb46fd616433d580cf2638693f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.496ex; height:3.343ex;" alt="{\displaystyle \textstyle {\binom {n}{0}},\;{\binom {n}{1}},\;\ldots ,\;{\binom {n}{n}}}"></span>: <ul><li>всі числа не кратні заданому простому <i>p</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></span> число <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> можна подати у вигляді <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mp^{k}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mp^{k}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3e4524f26190281add38aa53140a5dba1d29f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.301ex; height:3.009ex;" alt="{\displaystyle mp^{k}-1}"></span>, де натуральне число <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m<p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11029ce963f8b42eec561b13da6c2d5114e67ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.308ex; height:2.176ex;" alt="{\displaystyle m<p}"></span>;</li> <li>всі числа, крім першого й останнього, кратні даному простому <i>p</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=p^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4db3a256bd98498b5dc982b002df7a4d78a9600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.751ex; height:3.009ex;" alt="{\displaystyle n=p^{k}}"></span>;</li> <li>кількість непарних чисел дорівнює степеню двійки, показник якої дорівнює кількості одиниць у двійковому записі числа <i>n</i>;</li> <li>парних і непарних чисел не може бути порівну;</li> <li>кількість чисел, не кратних простому <i>p</i>, дорівнює <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}+1)*\dots *(a_{m}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗<!-- ∗ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}+1)*\dots *(a_{m}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d908e4741d6d9672ef2a4926ba1109f6062f159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.926ex; height:2.843ex;" alt="{\displaystyle (a_{1}+1)*\dots *(a_{m}+1)}"></span>, де числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\;\ldots ,a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\;\ldots ,a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7429a7677879534bbc811299fea2f2beda336512" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.012ex; height:2.009ex;" alt="{\displaystyle a_{1},\;\ldots ,a_{m}}"></span> — розряди <i>p</i>-кового запису числа <i>n</i>; а число <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m=\lfloor \log _{p}{n}\rfloor +1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle m=\lfloor \log _{p}{n}\rfloor +1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42778ff193c28abe2f11dd0889180f4c7af86989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.666ex; height:3.176ex;" alt="{\displaystyle \textstyle m=\lfloor \log _{p}{n}\rfloor +1,}"></span> де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \cdot \rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mo>⋅<!-- ⋅ --></mo> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \cdot \rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb5178c0fbe4ac275f1083ee8f1a6e0feb8f872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.712ex; height:2.843ex;" alt="{\displaystyle \lfloor \cdot \rfloor }"></span> — <a href="/wiki/%D0%A6%D1%96%D0%BB%D0%B0_%D1%87%D0%B0%D1%81%D1%82%D0%B8%D0%BD%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Ціла частина числа">функція «підлоги»</a> — це довжина даного запису.</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Тотожності"><span id=".D0.A2.D0.BE.D1.82.D0.BE.D0.B6.D0.BD.D0.BE.D1.81.D1.82.D1.96"></span>Тотожності</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=6" title="Редагувати розділ: Тотожності" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=6" title="Редагувати вихідний код розділу: Тотожності"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203b128a098e18cbb8cf36d004bd7282b28461bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.392ex; height:6.176ex;" alt="{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d21a79f459d51045422dcc73cc08b00fc07410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.016ex; height:6.176ex;" alt="{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}+{n \choose 1}+\cdots +{n \choose n}=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}+{n \choose 1}+\cdots +{n \choose n}=2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e5b33c706ec433e86d8e46e9d29d48fcf06ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.171ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}+{n \choose 1}+\cdots +{n \choose n}=2^{n}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}+{n \choose 2}+\cdots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}+{n \choose 2}+\cdots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f81a618ec3ad3e086447f88f55be07649e685cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.824ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}+{n \choose 2}+\cdots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\cdots +{n \choose n}^{2}={2n \choose n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\cdots +{n \choose n}^{2}={2n \choose n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfef119d96d15f8f9b28fd0c7d455e36e1df97c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.932ex; height:6.509ex;" alt="{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\cdots +{n \choose n}^{2}={2n \choose n}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>r</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>s</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>r</mi> <mo>+</mo> <mi>s</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8784fa7545285b7b90f6f8da0e6a36def0f1f854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.918ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}"></span> (згортка <a href="/wiki/%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80-%D0%A2%D0%B5%D0%BE%D1%84%D1%96%D0%BB_%D0%92%D0%B0%D0%BD%D0%B4%D0%B5%D1%80%D0%BC%D0%BE%D0%BD%D0%B4" title="Александр-Теофіл Вандермонд">Вандермонда</a>)</li></ol> <div class="mw-heading mw-heading3"><h3 id="Біном_Ньютона_і_наслідки"><span id=".D0.91.D1.96.D0.BD.D0.BE.D0.BC_.D0.9D.D1.8C.D1.8E.D1.82.D0.BE.D0.BD.D0.B0_.D1.96_.D0.BD.D0.B0.D1.81.D0.BB.D1.96.D0.B4.D0.BA.D0.B8"></span>Біном Ньютона і наслідки</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=7" title="Редагувати розділ: Біном Ньютона і наслідки" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=7" title="Редагувати вихідний код розділу: Біном Ньютона і наслідки"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}+{n \choose 1}+\ldots +{n \choose n}=2^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}+{n \choose 1}+\ldots +{n \choose n}=2^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6bf675292f7593bde557062ce7dbed7f3b9ddd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.818ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}+{n \choose 1}+\ldots +{n \choose n}=2^{n},}"></span> де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b82f4024a32a795b10f00971e03ad88ebaa367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i+j=m}{\binom {n}{j}}{\binom {n}{i}}(-1)^{j}={\begin{cases}{\binom {n}{m/2}},&{\text{якщо}}\ m\equiv 0{\pmod {2}},\\0,&{\text{якщо}}\ m\equiv 1{\pmod {2}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>m</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>якщо</mtext> </mrow> <mtext> </mtext> <mi>m</mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>якщо</mtext> </mrow> <mtext> </mtext> <mi>m</mi> <mo>≡<!-- ≡ --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i+j=m}{\binom {n}{j}}{\binom {n}{i}}(-1)^{j}={\begin{cases}{\binom {n}{m/2}},&{\text{якщо}}\ m\equiv 0{\pmod {2}},\\0,&{\text{якщо}}\ m\equiv 1{\pmod {2}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f488ffed7bdcf562169b42f016726f06da5d0dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.614ex; height:7.676ex;" alt="{\displaystyle \sum _{i+j=m}{\binom {n}{j}}{\binom {n}{i}}(-1)^{j}={\begin{cases}{\binom {n}{m/2}},&{\text{якщо}}\ m\equiv 0{\pmod {2}},\\0,&{\text{якщо}}\ m\equiv 1{\pmod {2}}.\end{cases}}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=k}^{n}{\binom {n}{j}}(-1)^{j}=(-1)^{k}{\binom {n-1}{k-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=k}^{n}{\binom {n}{j}}(-1)^{j}=(-1)^{k}{\binom {n-1}{k-1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f0d7301eb24fc209f69f99f1262b73ce38a5bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.68ex; height:7.176ex;" alt="{\displaystyle \sum _{j=k}^{n}{\binom {n}{j}}(-1)^{j}=(-1)^{k}{\binom {n-1}{k-1}}.}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}-{n \choose 1}+\ldots +(-1)^{n}{n \choose n}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}-{n \choose 1}+\ldots +(-1)^{n}{n \choose n}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4e0783976619a97a52dc1d5a35a004717127a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.598ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}-{n \choose 1}+\ldots +(-1)^{n}{n \choose n}=0,}"></span> де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b82f4024a32a795b10f00971e03ad88ebaa367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}"></span></li> <li>Сильніша тотожність: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}+{n \choose 2}+\ldots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}+{n \choose 2}+\ldots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a43c4597e282942c65e115e6aae97109bc076bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.471ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}+{n \choose 2}+\ldots +{n \choose 2\lfloor n/2\rfloor }=2^{n-1},}"></span> де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b82f4024a32a795b10f00971e03ad88ebaa367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{2a \choose k+a}^{3}={\frac {(3a)!}{(a!)^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mi>a</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{2a \choose k+a}^{3}={\frac {(3a)!}{(a!)^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3429d4fbfe928790641c40e735aa85b70b4555c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.338ex; height:7.176ex;" alt="{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{2a \choose k+a}^{3}={\frac {(3a)!}{(a!)^{3}}},}"></span></li></ul> <p>а в загальнішому вигляді </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}={\frac {(a+b+c)!}{a!\,b!\,c!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>c</mi> <mo>+</mo> <mi>a</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mi>a</mi> <mo>!</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo>!</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}={\frac {(a+b+c)!}{a!\,b!\,c!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010847eba30ce6bc85ed1c9f5e509597732280d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.776ex; height:7.009ex;" alt="{\displaystyle \sum _{k=-a}^{a}(-1)^{k}{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}={\frac {(a+b+c)!}{a!\,b!\,c!}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Згортка_Вандермонда_і_наслідки"><span id=".D0.97.D0.B3.D0.BE.D1.80.D1.82.D0.BA.D0.B0_.D0.92.D0.B0.D0.BD.D0.B4.D0.B5.D1.80.D0.BC.D0.BE.D0.BD.D0.B4.D0.B0_.D1.96_.D0.BD.D0.B0.D1.81.D0.BB.D1.96.D0.B4.D0.BA.D0.B8"></span>Згортка Вандермонда і наслідки</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=8" title="Редагувати розділ: Згортка Вандермонда і наслідки" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=8" title="Редагувати вихідний код розділу: Згортка Вандермонда і наслідки"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>r</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>s</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>r</mi> <mo>+</mo> <mi>s</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67416acc029d19f11f78289df9d44b7e34254be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.918ex; height:6.676ex;" alt="{\displaystyle \sum _{k}{r \choose m+k}{s \choose n-k}={r+s \choose m+n}}"></span> (<b>згортка <a href="/wiki/%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80-%D0%A2%D0%B5%D0%BE%D1%84%D1%96%D0%BB_%D0%92%D0%B0%D0%BD%D0%B4%D0%B5%D1%80%D0%BC%D0%BE%D0%BD%D0%B4" title="Александр-Теофіл Вандермонд">Вандермонда</a></b>), де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a017e129f6a1a45907f41e7808e3acdef0309e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.507ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {Z} ,}"></span> а <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,s\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,s\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22d73f98e1fef615ea52f600c7ae4111f8a079f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.339ex; height:2.509ex;" alt="{\displaystyle r,s\in \mathbb {R} .}"></span></li></ul> <p>Це тотожність виходить обчисленням коефіцієнта при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{m+n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{m+n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287e5075f380bf8b5018a59d533fb9f2c42b31db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.269ex; height:2.509ex;" alt="{\displaystyle x^{m+n}}"></span> у розкладі <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{r}(1+x)^{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{r}(1+x)^{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda96526e0d824a16521dc4b9f5cf1f8cf84f91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.261ex; height:2.843ex;" alt="{\displaystyle (1+x)^{r}(1+x)^{s}}"></span> з урахуванням тотожності <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{r+s}=(1+x)^{r}(1+x)^{s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>+</mo> <mi>s</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{r+s}=(1+x)^{r}(1+x)^{s}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbefbf8c649dfc534fd71cee81fef00a1acb108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.171ex; height:3.009ex;" alt="{\displaystyle (1+x)^{r+s}=(1+x)^{r}(1+x)^{s}.}"></span> Сума береться за всіма цілими <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, для яких <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {r \choose m+k}{s \choose n-k}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>r</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>s</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>≠<!-- ≠ --></mo> <mn>0.</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {r \choose m+k}{s \choose n-k}\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e0780a84f80914834d3bb960cc87c1b452e664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.866ex; height:3.343ex;" alt="{\displaystyle \textstyle {r \choose m+k}{s \choose n-k}\neq 0.}"></span> Для довільних дійсних <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> число ненульових доданків у сумі буде скінченним. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}{a \choose a}-{n \choose 1}{a+1 \choose a}+\ldots +(-1)^{n}{n \choose n}{a+n \choose a}=(-1)^{n}{a \choose n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>a</mi> <mi>a</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>a</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>a</mi> <mo>+</mo> <mi>n</mi> </mrow> <mi>a</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>a</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}{a \choose a}-{n \choose 1}{a+1 \choose a}+\ldots +(-1)^{n}{n \choose n}{a+n \choose a}=(-1)^{n}{a \choose n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1cb66fc4c917698e9453ddc80b113db09d705a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.794ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}{a \choose a}-{n \choose 1}{a+1 \choose a}+\ldots +(-1)^{n}{n \choose n}{a+n \choose a}=(-1)^{n}{a \choose n}}"></span>.</li> <li>загальніша тотожність: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{p}(-1)^{i}{p \choose i}\prod _{m=1}^{n}{i+s_{m} \choose s_{m}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{p}(-1)^{i}{p \choose i}\prod _{m=1}^{n}{i+s_{m} \choose s_{m}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24235be8d4bc523fe4a1664917621b754cbac564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.933ex; height:7.009ex;" alt="{\displaystyle \sum _{i=0}^{p}(-1)^{i}{p \choose i}\prod _{m=1}^{n}{i+s_{m} \choose s_{m}}=0}"></span>, якщо <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{m=1}^{n}{s_{m}}<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{n}{s_{m}}<p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c11e3bc9fe44589586cc4526ddc4ded346af207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.964ex; height:6.843ex;" alt="{\displaystyle \sum _{m=1}^{n}{s_{m}}<p}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\ldots +{n \choose n}^{2}={2n \choose n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\ldots +{n \choose n}^{2}={2n \choose n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97c6ed7fce6ad1e103166720b62ba5b4ee1015a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.578ex; height:6.509ex;" alt="{\displaystyle {n \choose 0}^{2}+{n \choose 1}^{2}+\ldots +{n \choose n}^{2}={2n \choose n}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Інші_тотожності"><span id=".D0.86.D0.BD.D1.88.D1.96_.D1.82.D0.BE.D1.82.D0.BE.D0.B6.D0.BD.D0.BE.D1.81.D1.82.D1.96"></span>Інші тотожності</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=9" title="Редагувати розділ: Інші тотожності" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=9" title="Редагувати вихідний код розділу: Інші тотожності"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\frac {(-1)^{k-1}}{k}}{n \choose k}=\sum _{k=1}^{n}{\frac {1}{k}}=H_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {(-1)^{k-1}}{k}}{n \choose k}=\sum _{k=1}^{n}{\frac {1}{k}}=H_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83373abac668735c4cd7551151363ecd1dbc5e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.499ex; height:7.009ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {(-1)^{k-1}}{k}}{n \choose k}=\sum _{k=1}^{n}{\frac {1}{k}}=H_{n}}"></span> — <i>n</i>- е <a href="/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D1%96%D1%87%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Гармонічне число">гармонічне число</a>.</li> <li><a href="/wiki/%D0%9C%D1%83%D0%BB%D1%8C%D1%82%D0%B8%D1%81%D0%B5%D0%BA%D1%86%D1%96%D1%8F_%D1%80%D1%8F%D0%B4%D1%83" title="Мультисекція ряду">Мультисекція ряду</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04845cc4a998213c15d61b36512d27222fca3527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle (1+x)^{n}}"></span> дає тотожність, що виражає суму біноміальних коефіцієнтів із довільним кроком <i>s</i> і зміщенням <i>t</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\leqslant t<s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⩽<!-- ⩽ --></mo> <mi>t</mi> <mo><</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0\leqslant t<s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb67e3aa30cc95f325d2aba9ef86e92d4019536c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.099ex; height:2.843ex;" alt="{\displaystyle (0\leqslant t<s)}"></span> у вигляді скінченної суми з <i>s</i> доданків:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{t}}+{\binom {n}{t+s}}+{\binom {n}{t+2s}}+\ldots ={\frac {1}{s}}\sum _{j=0}^{s-1}\left(2\cos {\frac {\pi j}{s}}\right)^{n}\cos {\frac {\pi (n-2t)j}{s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>t</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>t</mi> <mo>+</mo> <mi>s</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>…<!-- … --></mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>cos</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π<!-- π --></mi> <mi>j</mi> </mrow> <mi>s</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π<!-- π --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mi>j</mi> </mrow> <mi>s</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{t}}+{\binom {n}{t+s}}+{\binom {n}{t+2s}}+\ldots ={\frac {1}{s}}\sum _{j=0}^{s-1}\left(2\cos {\frac {\pi j}{s}}\right)^{n}\cos {\frac {\pi (n-2t)j}{s}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142a3037f965aeac48a9d348c405495b26b59fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:71.351ex; height:7.676ex;" alt="{\displaystyle {\binom {n}{t}}+{\binom {n}{t+s}}+{\binom {n}{t+2s}}+\ldots ={\frac {1}{s}}\sum _{j=0}^{s-1}\left(2\cos {\frac {\pi j}{s}}\right)^{n}\cos {\frac {\pi (n-2t)j}{s}}.}"></span></dd></dl> <ul><li>Виконуються рівності<sup id="cite_ref-example_1-0" class="reference"><a href="#cite_note-example-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}{\binom {n}{3}}&={\frac {n(n-1)(n-2)}{\color {Green}2}}-\sum _{i=2}^{n-1}{(n-i)(n-i+1)}=\\&=n(n-1)(n-2)-\sum _{i=2}^{n-1}{(n-i)({\color {Green}2}n-i+1)}=\\&=3{\binom {n}{3}}-2{\binom {n}{3}};\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mstyle mathcolor="#55ff55"> <mn>2</mn> </mstyle> </mfrac> </mrow> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>2</mn> </mstyle> </mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\binom {n}{3}}&={\frac {n(n-1)(n-2)}{\color {Green}2}}-\sum _{i=2}^{n-1}{(n-i)(n-i+1)}=\\&=n(n-1)(n-2)-\sum _{i=2}^{n-1}{(n-i)({\color {Green}2}n-i+1)}=\\&=3{\binom {n}{3}}-2{\binom {n}{3}};\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490f7bcbda98ccf1ed65ffd92544aa45c0acde72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:52.369ex; height:21.176ex;" alt="{\displaystyle {\begin{alignedat}{2}{\binom {n}{3}}&={\frac {n(n-1)(n-2)}{\color {Green}2}}-\sum _{i=2}^{n-1}{(n-i)(n-i+1)}=\\&=n(n-1)(n-2)-\sum _{i=2}^{n-1}{(n-i)({\color {Green}2}n-i+1)}=\\&=3{\binom {n}{3}}-2{\binom {n}{3}};\\\end{alignedat}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}{\binom {n}{4}}&={\frac {n(n-1)(n-2)(n-3)}{\color {Green}2}}-\sum _{i=3}^{n-1}{(n-i)\left(n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=n(n-1)(n-2)(n-3)-\sum _{i=3}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=24{\binom {n}{4}}-23{\binom {n}{4}};\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mstyle mathcolor="#00A64F"> <mn>2</mn> </mstyle> </mfrac> </mrow> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#00A64F"> <mn>2</mn> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>24</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−<!-- − --></mo> <mn>23</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\binom {n}{4}}&={\frac {n(n-1)(n-2)(n-3)}{\color {Green}2}}-\sum _{i=3}^{n-1}{(n-i)\left(n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=n(n-1)(n-2)(n-3)-\sum _{i=3}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=24{\binom {n}{4}}-23{\binom {n}{4}};\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc78c2b9c699bc56b64c5f59a38e08be7ac6f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:69.98ex; height:21.843ex;" alt="{\displaystyle {\begin{alignedat}{2}{\binom {n}{4}}&={\frac {n(n-1)(n-2)(n-3)}{\color {Green}2}}-\sum _{i=3}^{n-1}{(n-i)\left(n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=n(n-1)(n-2)(n-3)-\sum _{i=3}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)-\sum _{i_{0}=1}^{i-2}i_{0}\right)}=\\&=24{\binom {n}{4}}-23{\binom {n}{4}};\\\end{alignedat}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}{\binom {n}{5}}&={\frac {n(n-1)(n-2)(n-3)(n-4)}{\color {Green}2}}-\\&-\sum _{i=4}^{n-1}{(n-i)\left(n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=n(n-1)(n-2)(n-3)(n-4)-\\&-\sum _{i=4}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=120{\binom {n}{5}}-119{\binom {n}{5}}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mstyle mathcolor="#00A64F"> <mn>2</mn> </mstyle> </mfrac> </mrow> <mo>−<!-- − --></mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#00A64F"> <mn>2</mn> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>120</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−<!-- − --></mo> <mn>119</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\binom {n}{5}}&={\frac {n(n-1)(n-2)(n-3)(n-4)}{\color {Green}2}}-\\&-\sum _{i=4}^{n-1}{(n-i)\left(n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=n(n-1)(n-2)(n-3)(n-4)-\\&-\sum _{i=4}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=120{\binom {n}{5}}-119{\binom {n}{5}}.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5edcc450ec84933bca0532f0f49a4c8444eb09cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.95ex; margin-bottom: -0.222ex; width:54.96ex; height:31.509ex;" alt="{\displaystyle {\begin{alignedat}{2}{\binom {n}{5}}&={\frac {n(n-1)(n-2)(n-3)(n-4)}{\color {Green}2}}-\\&-\sum _{i=4}^{n-1}{(n-i)\left(n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=n(n-1)(n-2)(n-3)(n-4)-\\&-\sum _{i=4}^{n-1}{(n-i)\left({\color {Green}2}n(n-1)(n-2)-\sum _{i_{0}=1}^{i-3}\sum _{i_{1}=1}^{i_{0}}i_{1}\right)}=\\&=120{\binom {n}{5}}-119{\binom {n}{5}}.\end{alignedat}}}"></span></dd></dl> <p>Звідки випливає: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{3}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)(2n-i+1)}}{2}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)\left(2A_{n}^{1}-{\binom {i-1}{1}}\right)}}{2}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{3}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)(2n-i+1)}}{2}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)\left(2A_{n}^{1}-{\binom {i-1}{1}}\right)}}{2}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefaa9728e502c6084a4369784075ee9b480b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.265ex; height:9.843ex;" alt="{\displaystyle {\binom {n}{3}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)(2n-i+1)}}{2}}={\frac {\sum \limits _{i=2}^{n-1}{(n-i)\left(2A_{n}^{1}-{\binom {i-1}{1}}\right)}}{2}};}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{4}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2n(n-1)-\sum \limits _{i_{0}=1}^{i-2}i_{0}\right)}}{23}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2A_{n}^{2}-{\binom {i-1}{2}}\right)}}{23}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mn>23</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mn>23</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{4}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2n(n-1)-\sum \limits _{i_{0}=1}^{i-2}i_{0}\right)}}{23}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2A_{n}^{2}-{\binom {i-1}{2}}\right)}}{23}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e0062a8c333a5a6bd74aacbca9cacbd731c7a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.669ex; height:11.009ex;" alt="{\displaystyle {\binom {n}{4}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2n(n-1)-\sum \limits _{i_{0}=1}^{i-2}i_{0}\right)}}{23}}={\frac {\sum \limits _{i=3}^{n-1}{(n-i)\left(2A_{n}^{2}-{\binom {i-1}{2}}\right)}}{23}};}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}&{\binom {n}{5}}={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2n(n-1)(n-2)-\sum \limits _{i_{0}=1}^{i-3}\sum \limits _{i_{1}=1}^{i_{0}}i_{1}\right)}}{119}}=\\&={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2A_{n}^{3}-{\binom {i-1}{3}}\right)}}{119}};\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mn>119</mn> </mfrac> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mn>119</mn> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}&{\binom {n}{5}}={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2n(n-1)(n-2)-\sum \limits _{i_{0}=1}^{i-3}\sum \limits _{i_{1}=1}^{i_{0}}i_{1}\right)}}{119}}=\\&={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2A_{n}^{3}-{\binom {i-1}{3}}\right)}}{119}};\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f48d3066203bdaddc46a8d4236676dc78c7d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:55.786ex; height:20.176ex;" alt="{\displaystyle {\begin{alignedat}{2}&{\binom {n}{5}}={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2n(n-1)(n-2)-\sum \limits _{i_{0}=1}^{i-3}\sum \limits _{i_{1}=1}^{i_{0}}i_{1}\right)}}{119}}=\\&={\frac {\sum \limits _{i=4}^{n-1}{(n-i)\left(2A_{n}^{3}-{\binom {i-1}{3}}\right)}}{119}};\\\end{alignedat}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\frac {\sum \limits _{i=k-1}^{n-1}{(n-i)\left(2A_{n}^{k-2}-{\binom {i-1}{k-2}}\right)}}{k!-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {\sum \limits _{i=k-1}^{n-1}{(n-i)\left(2A_{n}^{k-2}-{\binom {i-1}{k-2}}\right)}}{k!-1}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db392e386963fe3af235aa6d50d7fbd3257bef52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.973ex; height:10.009ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {\sum \limits _{i=k-1}^{n-1}{(n-i)\left(2A_{n}^{k-2}-{\binom {i-1}{k-2}}\right)}}{k!-1}},}"></span></dd></dl> <p>де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048c75440ffaa800c55a86a58042722bedec0101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.843ex;" alt="{\displaystyle A_{n}^{k}}"></span> — кількість <a href="/wiki/%D0%A0%D0%BE%D0%B7%D0%BC%D1%96%D1%89%D0%B5%D0%BD%D0%BD%D1%8F_(%D0%BA%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0)" title="Розміщення (комбінаторика)">розміщень</a> із <i>n</i> по <i>k</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Матричні_співвідношення"><span id=".D0.9C.D0.B0.D1.82.D1.80.D0.B8.D1.87.D0.BD.D1.96_.D1.81.D0.BF.D1.96.D0.B2.D0.B2.D1.96.D0.B4.D0.BD.D0.BE.D1.88.D0.B5.D0.BD.D0.BD.D1.8F"></span>Матричні співвідношення</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=10" title="Редагувати розділ: Матричні співвідношення" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=10" title="Редагувати вихідний код розділу: Матричні співвідношення"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Якщо взяти квадратну матрицю, відрахувавши <i>N</i> елементів по катетах трикутника Паскаля і повернувши матрицю на будь-який з чотирьох кутів, то <a href="/wiki/%D0%92%D0%B8%D0%B7%D0%BD%D0%B0%D1%87%D0%BD%D0%B8%D0%BA" title="Визначник">детермінант</a> цих чотирьох матриць дорівнює ±1 за будь-якого <i>N</i>, причому детермінант матриці з вершиною трикутника у верхньому лівому куті дорівнює 1. </p><p>У матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd247ede4214d444729cf39d5cd1201f6232f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.344ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"></span> числа на діагоналі <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i+j=const}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i+j=const}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96429f947e43ccc99b0d8fd0fd696012ad9e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.159ex; height:2.509ex;" alt="{\displaystyle i+j=const}"></span> повторюють числа рядків трикутника Паскаля <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j=0,1,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j=0,1,\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d960d86b9c738b6040f570c519d5d832fe310be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle (i,j=0,1,\ldots )}"></span>. Її можна розкласти в добуток двох строго діагональних матриць: нижньотрикутної та одержуваної з неї транспонуванням. А саме: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}=UU^{T},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>U</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}=UU^{T},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e82934ed6ed9739f6fc353eb7df77989bb59ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.102ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}=UU^{T},}"></span></dd></dl> <p>де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\begin{bmatrix}{\tbinom {i}{j}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>i</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\begin{bmatrix}{\tbinom {i}{j}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b36480c6c38d6fcf64470325da7d9193c91bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.635ex; height:4.843ex;" alt="{\displaystyle U={\begin{bmatrix}{\tbinom {i}{j}}\end{bmatrix}}}"></span>. Обернена матриця до <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> має вигляд: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{-1}={\begin{bmatrix}(-1)^{i+j}{\binom {i}{j}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>i</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{-1}={\begin{bmatrix}(-1)^{i+j}{\binom {i}{j}}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8884193f4a6f3f0dcd7574b17cd11f34cb1cb18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.208ex; height:4.843ex;" alt="{\displaystyle U^{-1}={\begin{bmatrix}(-1)^{i+j}{\binom {i}{j}}\end{bmatrix}}.}"></span></dd></dl> <p>Таким чином, матрицю, обернену до <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd247ede4214d444729cf39d5cd1201f6232f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.344ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"></span>, можна розкласти в добуток двох строго діагональних матриць: перша матриця — верхньотрикутна, а друга виходить з першої транспонуванням, що дозволяє дати явний вираз для обернених елементів: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}=\sum _{k=0}^{p}(-1)^{m+n}{\binom {k}{m}}{\binom {k}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}=\sum _{k=0}^{p}(-1)^{m+n}{\binom {k}{m}}{\binom {k}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915d7eee7514f9f168c3695ab589beda99afe7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.913ex; height:7.176ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}=\sum _{k=0}^{p}(-1)^{m+n}{\binom {k}{m}}{\binom {k}{n}}}"></span>, де <i>i</i>, <i>j</i>, <i>m</i>, <i>n</i> = 0..<i>p</i>.</dd></dl> <p>Елементи оберненої матриці змінюються за зміни її розміру і, на відміну від матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd247ede4214d444729cf39d5cd1201f6232f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.344ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"></span>, недостатньо приписати новий рядок і стовпець. Стовпець <i>j</i> матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd247ede4214d444729cf39d5cd1201f6232f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.344ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"></span> є многочленом степеня <i>j</i> за аргументом <i>i</i>, отже, перші p стовпців утворюють повний базис у просторі векторів довжини <i>p</i>+1, чиї координати можна інтерполювати многочленом степеня рівного або меншого ніж <i>p</i>-1. Нижній рядок матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a48401c12eaaf2f59474bfb2aa3c958a5bd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.463ex; height:4.343ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"></span> ортогональний до будь-якого такого вектора. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{p,n}^{-1}=\sum _{k=0}^{p}(-1)^{p+n}{\binom {k}{p}}{\binom {k}{n}}=(-1)^{p+n}{\binom {p}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>p</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{p,n}^{-1}=\sum _{k=0}^{p}(-1)^{p+n}{\binom {k}{p}}{\binom {k}{n}}=(-1)^{p+n}{\binom {p}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936923157729df9e1e349973d4f7c90b60181e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.871ex; height:7.176ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{p,n}^{-1}=\sum _{k=0}^{p}(-1)^{p+n}{\binom {k}{p}}{\binom {k}{n}}=(-1)^{p+n}{\binom {p}{n}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{P}_{a}(n)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{P}_{a}(n)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b1c5e409ef8f2cb2d138b15cfea2c22ea77f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.334ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{P}_{a}(n)=0}"></span> при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab2a8f9036aead7e07e0a0de824ef61684e2421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.498ex; height:2.176ex;" alt="{\displaystyle a<p}"></span>, де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {P}_{a}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {P}_{a}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8152a46558fbef5f4d47503e5ec7c1cc5fa0a7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.798ex; height:2.843ex;" alt="{\displaystyle {P}_{a}(n)}"></span> многочлен степеня <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>.</dd></dl> <p>Якщо довільний вектор довжини <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885ec01d3b5670fd5f88847f32da2b3dd62f60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p+1}"></span> можна інтерполювати многочленом степеня <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c42e6b0d030b83ae866a6cac9de7225f181f2fe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.07ex; height:2.509ex;" alt="{\displaystyle i<p}"></span>, то скалярний добуток з рядками <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i+1,i+2,\dots ,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i+1,i+2,\dots ,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375f14298cd10bebfd543ceb68ea0de85be36407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.992ex; height:2.509ex;" alt="{\displaystyle i+1,i+2,\dots ,p}"></span> (нумерація з 0) матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a48401c12eaaf2f59474bfb2aa3c958a5bd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.463ex; height:4.343ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"></span> дорівнює нулю. Використовуючи тотожність вище і рівність одиниці скалярного добутку нижнього рядка матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a48401c12eaaf2f59474bfb2aa3c958a5bd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.463ex; height:4.343ex;" alt="{\displaystyle {\begin{bmatrix}{\binom {i+j}{i}}\end{bmatrix}}_{m,n}^{-1}}"></span> на останній стовпець матриці <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd247ede4214d444729cf39d5cd1201f6232f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.344ex; height:3.509ex;" alt="{\displaystyle {\begin{bmatrix}{\tbinom {i+j}{i}}\end{bmatrix}}}"></span>, маємо: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p}=p!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mi>p</mi> <mo>!</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p}=p!.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a07508bf58204341d1b6919b57f42a1bc8f871c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.29ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p}=p!.}"></span></dd></dl> <p>Для показника, більшого від <i>p</i>, можна задати рекурентну формулу: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p+a}=p!{P}_{2a}(p)={f}_{a}(p),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>p</mi> <mo>!</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p+a}=p!{P}_{2a}(p)={f}_{a}(p),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318e62ef7949ddfdc75008dae6b092e4b6a5e8a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.151ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{p}(-1)^{p+n}{\binom {p}{n}}{n}^{p+a}=p!{P}_{2a}(p)={f}_{a}(p),}"></span></dd></dl> <p>де многочлен </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {P}_{2a+2}(p)=\sum _{x=1}^{p}x{P}_{2a}(x);\quad a\geqslant 0;\quad {P}_{0}(p)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mi>a</mi> <mo>⩾<!-- ⩾ --></mo> <mn>0</mn> <mo>;</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {P}_{2a+2}(p)=\sum _{x=1}^{p}x{P}_{2a}(x);\quad a\geqslant 0;\quad {P}_{0}(p)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163ca343c62a143ac12cbd916d6a0c7f11ab64c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.857ex; height:7.009ex;" alt="{\displaystyle {P}_{2a+2}(p)=\sum _{x=1}^{p}x{P}_{2a}(x);\quad a\geqslant 0;\quad {P}_{0}(p)=1.}"></span></dd></dl> <p>Для доведення спершу доводиться тотожність: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f}_{a}(p+1)=\sum _{x=0}^{a}{(p+1)}^{x+1}{f}_{a-x}(p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−<!-- − --></mo> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {f}_{a}(p+1)=\sum _{x=0}^{a}{(p+1)}^{x+1}{f}_{a-x}(p).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c4433b6ae3d4c5f5aecf7accfc74a498546ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.403ex; height:6.843ex;" alt="{\displaystyle {f}_{a}(p+1)=\sum _{x=0}^{a}{(p+1)}^{x+1}{f}_{a-x}(p).}"></span></dd></dl> <p>Якщо потрібно знайти формулу не для всіх показників степеня, то </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {P}_{2a}(p)={\frac {p}{{2}^{a}}}{\binom {p+a}{a}}{Q}_{a-1}(p);\quad a>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>p</mi> <mo>+</mo> <mi>a</mi> </mrow> <mi>a</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mi>a</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {P}_{2a}(p)={\frac {p}{{2}^{a}}}{\binom {p+a}{a}}{Q}_{a-1}(p);\quad a>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff68366e4e2fe168b3f2030a80547f098c583885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.768ex; height:6.176ex;" alt="{\displaystyle {P}_{2a}(p)={\frac {p}{{2}^{a}}}{\binom {p+a}{a}}{Q}_{a-1}(p);\quad a>0.}"></span></dd></dl> <p>Старший коефіцієнт <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Q}_{a-1}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Q}_{a-1}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b488d8ac5408e7690d165647988bd05322c41e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.019ex; height:3.009ex;" alt="{\displaystyle {Q}_{a-1}(p)}"></span> дорівнює 1, щоб знайти інші коефіцієнти, знадобиться <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240dcbc0ddbf5932d0ea301ef5576b46ba12d26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a-1}"></span> значень: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Q}_{a-1}(p)=p(p+1){T}_{a-3}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Q}_{a-1}(p)=p(p+1){T}_{a-3}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e37912bc090d1f2fa6e768c88c9a4f0fb2fb7484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.808ex; height:3.009ex;" alt="{\displaystyle {Q}_{a-1}(p)=p(p+1){T}_{a-3}(p)}"></span> для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv 1{\pmod {2}};a\geqslant 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡<!-- ≡ --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <mi>a</mi> <mo>⩾<!-- ⩾ --></mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv 1{\pmod {2}};a\geqslant 3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09155bdd2fef20bf8ec780b038c1c9baedc3d269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.509ex; height:2.843ex;" alt="{\displaystyle a\equiv 1{\pmod {2}};a\geqslant 3.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Цілозначні_многочлени"><span id=".D0.A6.D1.96.D0.BB.D0.BE.D0.B7.D0.BD.D0.B0.D1.87.D0.BD.D1.96_.D0.BC.D0.BD.D0.BE.D0.B3.D0.BE.D1.87.D0.BB.D0.B5.D0.BD.D0.B8"></span>Цілозначні многочлени</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=11" title="Редагувати розділ: Цілозначні многочлени" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=11" title="Редагувати вихідний код розділу: Цілозначні многочлени"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Біноміальні коефіцієнти <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {x}{0}}=1,{\tbinom {x}{1}}=x,{\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {x}{0}}=1,{\tbinom {x}{1}}=x,{\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba73e03ff77db9e1a9323f8dc3b5e9fcb670fe0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.937ex; height:4.009ex;" alt="{\displaystyle {\tbinom {x}{0}}=1,{\tbinom {x}{1}}=x,{\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}},}"></span> … є <i>цілозначними</i> <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен">многочленами</a> від <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, тобто, набувають цілих значень за цілих значень <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> — це неважко зрозуміти, наприклад, за трикутником Паскаля. Більш того, вони утворюють базис цілозначних многочленів, у якому всі цілозначні многочлени виражаються як <a href="/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%86%D1%96%D1%8F" title="Лінійна комбінація">лінійні комбінації</a> з цілими коефіцієнтами<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p><p>Разом з тим, стандартний базис <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,x,x^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,x,x^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68457c27f4e970f9da0909bd775d1f7c6816845e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.591ex; height:3.009ex;" alt="{\displaystyle 1,x,x^{2},}"></span> … не дозволяє виразити всіх цілочисельних многочленів, якщо використовувати тільки цілі коефіцієнти, оскільки вже <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8ec9af2d157aa25af5dea8dbb90cae8298bafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.393ex; height:4.009ex;" alt="{\displaystyle {\tbinom {x}{2}}={\tfrac {x^{2}}{2}}-{\tfrac {x}{2}}}"></span> має дробові коефіцієнти при степенях <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p><p>Цей результат узагальнюється на многочлени багатьох змінних. А саме, якщо многочлен <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{1},\dots ,x_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x_{1},\dots ,x_{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa611e0fe99b659558f5435491d143111e7a1e64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.14ex; height:2.843ex;" alt="{\displaystyle R(x_{1},\dots ,x_{m})}"></span> степеня <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> має дійсні коефіцієнти і набуває цілих значень за цілих значень змінних, то </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{1},\dots ,x_{m})=P\left({\binom {x_{1}}{1}},\dots ,{\binom {x_{1}}{k}},\dots ,{\binom {x_{m}}{1}},\dots ,{\binom {x_{m}}{k}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x_{1},\dots ,x_{m})=P\left({\binom {x_{1}}{1}},\dots ,{\binom {x_{1}}{k}},\dots ,{\binom {x_{m}}{1}},\dots ,{\binom {x_{m}}{k}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a6c59efb8cf1ff71727c332329447ba89f4273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.823ex; height:6.176ex;" alt="{\displaystyle R(x_{1},\dots ,x_{m})=P\left({\binom {x_{1}}{1}},\dots ,{\binom {x_{1}}{k}},\dots ,{\binom {x_{m}}{1}},\dots ,{\binom {x_{m}}{k}}\right),}"></span></dd></dl> <p>де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> — многочлен із цілими коефіцієнтами.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Асимптотика_і_оцінки"><span id=".D0.90.D1.81.D0.B8.D0.BC.D0.BF.D1.82.D0.BE.D1.82.D0.B8.D0.BA.D0.B0_.D1.96_.D0.BE.D1.86.D1.96.D0.BD.D0.BA.D0.B8"></span>Асимптотика і оцінки</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=12" title="Редагувати розділ: Асимптотика і оцінки" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=12" title="Редагувати вихідний код розділу: Асимптотика і оцінки"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {2n \choose n}\sim {\frac {2^{2n}}{\sqrt {\pi n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>∼<!-- ∼ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msqrt> <mi>π<!-- π --></mi> <mi>n</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2n \choose n}\sim {\frac {2^{2n}}{\sqrt {\pi n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea316f6a187dc2d17252440bdd0595e1984921e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.576ex; height:6.676ex;" alt="{\displaystyle {2n \choose n}\sim {\frac {2^{2n}}{\sqrt {\pi n}}}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq {\frac {n}{(n/2-m)^{2}}}2^{n-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>≤<!-- ≤ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−<!-- − --></mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq {\frac {n}{(n/2-m)^{2}}}2^{n-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c2a9ea90f3e6ad3c64c08d73aa37f29fd5b5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.438ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq {\frac {n}{(n/2-m)^{2}}}2^{n-3}}"></span> при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\leq n/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≤<!-- ≤ --></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\leq n/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54164a67468ec6dfb1348223e871504611d0240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle m\leq n/2}"></span> (<a href="/wiki/%D0%9D%D0%B5%D1%80%D1%96%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%A7%D0%B5%D0%B1%D0%B8%D1%88%D0%B5%D0%B2%D0%B0" class="mw-redirect" title="Нерівність Чебишева">нерівність Чебишева</a>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq 2^{nH(m/n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>≤<!-- ≤ --></mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq 2^{nH(m/n)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7738d7774930248d59df7b1b884212af037250e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.027ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{m}{n \choose k}\leq 2^{nH(m/n)}}"></span> (<a href="/w/index.php?title=%D0%95%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D1%96%D0%B9%D0%BD%D0%B0_%D0%BE%D1%86%D1%96%D0%BD%D0%BA%D0%B0&action=edit&redlink=1" class="new" title="Ентропійна оцінка (ще не написана)">ентропійна оцінка</a>), де <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=-x\log _{2}x-(1-x)\log _{2}(1-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mi>x</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)=-x\log _{2}x-(1-x)\log _{2}(1-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d138361f033797b87ec541817e6b9c836ed69b7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.106ex; height:2.843ex;" alt="{\displaystyle H(x)=-x\log _{2}x-(1-x)\log _{2}(1-x)}"></span> — <a href="/wiki/%D0%95%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D1%96%D1%8F" title="Ентропія">ентропія</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n/2-\lambda }{n \choose k}\leq 2^{n}e^{-2\lambda ^{2}/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>≤<!-- ≤ --></mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> <msup> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n/2-\lambda }{n \choose k}\leq 2^{n}e^{-2\lambda ^{2}/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69eead681edf4e72a4f87153a15458b83bc83881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.564ex; height:7.843ex;" alt="{\displaystyle \sum _{k=0}^{n/2-\lambda }{n \choose k}\leq 2^{n}e^{-2\lambda ^{2}/n}}"></span> (<a href="/wiki/%D0%9D%D0%B5%D1%80%D1%96%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%A7%D0%B5%D1%80%D0%BD%D0%BE%D0%B2%D0%B0" title="Нерівність Чернова">нерівність Чернова</a>)</li></ol> <div class="mw-heading mw-heading2"><h2 id="Алгоритми_обчислення"><span id=".D0.90.D0.BB.D0.B3.D0.BE.D1.80.D0.B8.D1.82.D0.BC.D0.B8_.D0.BE.D0.B1.D1.87.D0.B8.D1.81.D0.BB.D0.B5.D0.BD.D0.BD.D1.8F"></span>Алгоритми обчислення</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=13" title="Редагувати розділ: Алгоритми обчислення" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=13" title="Редагувати вихідний код розділу: Алгоритми обчислення"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Біноміальні коефіцієнти можуть бути обчислені за допомогою тотожності <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={n-1 \choose k}+{n-1 \choose k-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={n-1 \choose k}+{n-1 \choose k-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771ae699706c84ac767df08db61d05b72e7ef173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.392ex; height:6.176ex;" alt="{\displaystyle {n \choose k}={n-1 \choose k}+{n-1 \choose k-1}}"></span>, якщо на кожному кроці зберігати значення <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}"></span> для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586793723e46deb17a24975194d80362cd3261eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.241ex; height:2.509ex;" alt="{\displaystyle k=0,1,\dots ,n}"></span>. Цей алгоритм особливо ефективний, якщо необхідно отримати всі значення <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}"></span> при фіксованому <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. Алгоритм потребує <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f1e87cae1db33414fd2cefca598e265d635f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.558ex; height:2.843ex;" alt="{\displaystyle \ O(n)}"></span> пам'яті (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34550c5a4e2e282e9f78cc46f2b0b738de45f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.612ex; height:3.176ex;" alt="{\displaystyle \ O(n^{2})}"></span> для обчислення всієї таблиці) і <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34550c5a4e2e282e9f78cc46f2b0b738de45f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.612ex; height:3.176ex;" alt="{\displaystyle \ O(n^{2})}"></span> часу.</li></ul> <ul><li>Інший спосіб ґрунтується на тотожності <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d21a79f459d51045422dcc73cc08b00fc07410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.016ex; height:6.176ex;" alt="{\displaystyle {n \choose k}={\frac {n}{n-k}}{n-1 \choose k}}"></span>. Він дозволяє обчислити значення <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}"></span> при фіксованому <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. Алгоритм потребує <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd44330334a8299fe180fcdb2f35638b12a2068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.843ex;" alt="{\displaystyle \ O(1)}"></span> пам'яті і <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a45708c37e924280a686176e67329bfed34737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.374ex; height:2.843ex;" alt="{\displaystyle \ O(k)}"></span> часу.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Узагальнення"><span id=".D0.A3.D0.B7.D0.B0.D0.B3.D0.B0.D0.BB.D1.8C.D0.BD.D0.B5.D0.BD.D0.BD.D1.8F"></span>Узагальнення</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=14" title="Редагувати розділ: Узагальнення" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=14" title="Редагувати вихідний код розділу: Узагальнення"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Оскільки для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n+1\in \mathbb {N} :\ n!=\Gamma (n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>:</mo> <mtext> </mtext> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n+1\in \mathbb {N} :\ n!=\Gamma (n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10df8860f760d9359d82adebc43fe259cb58d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.814ex; height:2.843ex;" alt="{\displaystyle \ n+1\in \mathbb {N} :\ n!=\Gamma (n+1)}"></span>, то значення біноміального коефіцієнта можна визначити для усіх комплексних чисел <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221cbaeced84c938e33968eb6511747913a2d3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.176ex;" alt="{\displaystyle \ n\in \mathbb {C} }"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d13dcd5dab75c540bc98da17d2b11b801afe62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.311ex; height:2.176ex;" alt="{\displaystyle \ k\in \mathbb {C} }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={\frac {\Gamma (n+1)}{\Gamma (k+1)\Gamma (n-k+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={\frac {\Gamma (n+1)}{\Gamma (k+1)\Gamma (n-k+1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/956efc1d92b1c4f724827451ca37887933fd2ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.938ex; height:6.509ex;" alt="{\displaystyle {n \choose k}={\frac {\Gamma (n+1)}{\Gamma (k+1)\Gamma (n-k+1)}}}"></span></dd></dl> <p>Явні формули для обчислення біноміальних коефіцієнтів для цілих чисел <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={\frac {n!}{k!\;(n-k)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={\frac {n!}{k!\;(n-k)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f2b110e4436c2b8a67256f0b3b7a00eb5df56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.156ex; height:6.343ex;" alt="{\displaystyle {n \choose k}={\frac {n!}{k!\;(n-k)!}}}"></span> для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo>≤<!-- ≤ --></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq k\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b429c3c44b3dc10332285272eee6f754dbf985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq k\leq n}"></span>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8ea4e63baa4433398ac0541a21354c5750ab76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.077ex; height:6.176ex;" alt="{\displaystyle {n \choose k}=0}"></span> для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9eddb74a10afb2f73144ab9294d18cae14db75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle \ k<0}"></span> або <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq n<k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>n</mi> <mo><</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq n<k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/621cfa9d30e64d963d053a04853386c488c09342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq n<k}"></span>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}=(-1)^{k}{-n+k-1 \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−<!-- − --></mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}=(-1)^{k}{-n+k-1 \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a863d97784d13badb4dc12b6042f5113602f98a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.461ex; height:6.176ex;" alt="{\displaystyle {n \choose k}=(-1)^{k}{-n+k-1 \choose k}}"></span> для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<0\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<0\leq k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916cf9bf99c4acdcbc8802c5102c04a1b0a5a311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle n<0\leq k}"></span>.</dd></dl> <p>Біноміальні коефіцієнти часто зустрічаються в <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбінаторика">комбінаторних</a> задачах і <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B9%D0%BC%D0%BE%D0%B2%D1%96%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Теорія ймовірностей">теорії імовірностей</a>. </p><p>Узагальненням біноміального коефіцієнта є <a href="/wiki/%D0%9C%D1%83%D0%BB%D1%8C%D1%82%D0%B8%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82" class="mw-redirect" title="Мультиноміальний коефіцієнт">мультиноміальний коефіцієнт</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Генерація_на_C++"><span id=".D0.93.D0.B5.D0.BD.D0.B5.D1.80.D0.B0.D1.86.D1.96.D1.8F_.D0.BD.D0.B0_C.2B.2B"></span>Генерація на C++</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=15" title="Редагувати розділ: Генерація на C++" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=15" title="Редагувати вихідний код розділу: Генерація на C++"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><iostream></span> <span class="cp">#include</span><span class="w"> </span><span class="cpf"><string></span> <span class="k">using</span><span class="w"> </span><span class="k">namespace</span><span class="w"> </span><span class="nn">std</span><span class="p">;</span> <span class="kt">void</span><span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">startAt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">""</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">startAt</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">m</span><span class="p">)</span> <span class="w"> </span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="p">)(</span><span class="n">i</span><span class="o">+</span><span class="sc">'0'</span><span class="p">)</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span> <span class="w"> </span><span class="k">else</span> <span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="p">)(</span><span class="n">i</span><span class="o">+</span><span class="sc">'0'</span><span class="p">));</span> <span class="w"> </span><span class="p">}</span><span class="w"> </span> <span class="p">}</span> <span class="kt">int</span><span class="w"> </span><span class="n">main</span><span class="p">()</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">);</span> <span class="w"> </span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span> <span class="p">}</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Див._також"><span id=".D0.94.D0.B8.D0.B2._.D1.82.D0.B0.D0.BA.D0.BE.D0.B6"></span>Див. також</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=16" title="Редагувати розділ: Див. також" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=16" title="Редагувати вихідний код розділу: Див. також"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%91%D1%96%D0%BD%D0%BE%D0%BC" title="Біном">Біном</a></li> <li><a href="/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81%D0%BE%D0%B2%D1%96_%D0%B1%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82%D0%B8" title="Гауссові біноміальні коефіцієнти">Гауссові біноміальні коефіцієнти</a></li> <li><a href="/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%B1%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82" title="Центральний біноміальний коефіцієнт">Центральний біноміальний коефіцієнт</a></li> <li><a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_fusc" title="Функція fusc">Функція fusc</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Примітки"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D1.96.D1.82.D0.BA.D0.B8"></span>Примітки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=17" title="Редагувати розділ: Примітки" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=17" title="Редагувати вихідний код розділу: Примітки"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r43816068">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist references-column-count references-column-count-2" style="column-count: 2; -moz-column-count: 2; -webkit-column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-example-1"><span class="mw-cite-backlink"><a href="#cite_ref-example_1-0">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r43245077">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ref-lang{font-size:85%;cursor:help;margin-left:0.2em;color:var(--color-subtle,#54595d)}.mw-parser-output .cs1-ref-lg{font-style:normal;cursor:help}.mw-parser-output .cs1-ref-lg-text{color:#252525;text-decoration:inherit;text-decoration-color:#252525}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-night .mw-parser-output .cs1-ref-lg-text{color:#dadad6;text-decoration-color:#dadad6}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-night .mw-parser-output .cs1-ref-lg-text{color:#dadad6;text-decoration-color:#dadad6}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20211025234309/https://habr.com/ru/post/530098/">Четырёхмерные таблицы в комбинаторике — два странных способа посчитать сочетания</a>. <a href="/wiki/Habr.com" class="mw-redirect" title="Habr.com">habr.com</a>. 30 листопада 2020. Архів <a rel="nofollow" class="external text" href="https://habr.com/ru/post/530098/">оригіналу</a> за 25 жовтня 2021<span class="reference-accessdate">. Процитовано 27 березня 2021</span>.</cite></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation"><i>Прасолов В. В.</i> Глава 12. Целозначные многочлены // <a rel="nofollow" class="external autonumber" href="http://inis.jinr.ru/sl/vol2/Mathematics/Алгебра/Прасолов,_Многочлены,2001.pdf">[1]</a> — М. : <a href="/wiki/%D0%9C%D0%A6%D0%9D%D0%9C%D0%9E" class="mw-redirect" title="МЦНМО">МЦНМО</a>, 1999, 2001, 2003. <small><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220121053620/http://inis.jinr.ru/sl/vol2/Mathematics/Алгебра/Прасолов,_Многочлены,2001.pdf">Архівовано</a> з джерела 21 січня 2022</small></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation"><i>Ю. Матиясевич</i>. Десятая проблема Гильберта. — Наука, 1993. — С. 20,185,194.</span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Джерела"><span id=".D0.94.D0.B6.D0.B5.D1.80.D0.B5.D0.BB.D0.B0"></span>Джерела</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&veaction=edit&section=18" title="Редагувати розділ: Джерела" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=edit&section=18" title="Редагувати вихідний код розділу: Джерела"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r43245077"><cite class="citation book cs1"><a href="/wiki/%D0%97%D0%B0%D0%B2%D0%B0%D0%BB%D0%BE_%D0%A1%D0%B5%D1%80%D0%B3%D1%96%D0%B9_%D0%A2%D1%80%D0%BE%D1%85%D0%B8%D0%BC%D0%BE%D0%B2%D0%B8%D1%87" title="Завало Сергій Трохимович">Завало С. Т.</a> (1972). <i>Елементи аналізу. Алгебра многочленів</i>. <a href="/wiki/%D0%9A%D0%B8%D1%97%D0%B2" title="Київ">Київ</a>: Радянська школа. с. 462.</cite> <style data-mw-deduplicate="TemplateStyles:r43693355">.mw-parser-output .ref-info{font-size:85%;cursor:help;margin-left:0.2em;color:var(--color-subtle,#54595d)}</style><span title="українською мовою" class="ref-info">(укр.)</span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r43815798">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output 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title="Обговорення сторінки [t]" accesskey="t"><span>Обговорення</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">українська</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Перегляди</span> </h3> <div class="vector-menu-content"> <ul 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href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=history" title="Журнал змін сторінки [h]" accesskey="h"><span>Переглянути історію</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Більше опцій" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Більше</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Пошук</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Пошук у Вікіпедії" aria-label="Пошук у Вікіпедії" autocapitalize="sentences" title="Шукати у Вікіпедії [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Спеціальна:Пошук"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Знайти сторінки, що містять зазначений текст" value="Знайти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти до сторінки, що має точно таку назву (якщо вона існує)" value="Перейти"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Перейти на головну сторінку"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Навігація</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Перейти на головну сторінку [z]" accesskey="z"><span>Головна сторінка</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB:%D0%9F%D0%BE%D1%82%D0%BE%D1%87%D0%BD%D1%96_%D0%BF%D0%BE%D0%B4%D1%96%D1%97" title="Список поточних подій"><span>Поточні події</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Список останніх змін у цій вікі [r]" accesskey="r"><span>Нові редагування</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8"><span>Нові сторінки</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Переглянути випадкову сторінку [x]" accesskey="x"><span>Випадкова стаття</span></a></li> </ul> </div> </nav> <nav id="p-Участь" class="mw-portlet mw-portlet-Участь vector-menu-portal portal vector-menu" aria-labelledby="p-Участь-label" > <h3 id="p-Участь-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Участь</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB_%D1%81%D0%BF%D1%96%D0%BB%D1%8C%D0%BD%D0%BE%D1%82%D0%B8" title="Про проєкт, про те, що Ви можете зробити, і що де шукати"><span>Портал спільноти</span></a></li><li id="n-tavern" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9A%D0%BD%D0%B0%D0%B9%D0%BF%D0%B0" title="Місце для обговорення більшості питань"><span>Кнайпа</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%94%D0%BE%D0%B2%D1%96%D0%B4%D0%BA%D0%B0" title="Довідка з проєкту"><span>Довідка</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_uk.wikipedia.org&uselang=uk" title="Підтримайте проєкт"><span>Пожертвувати</span></a></li><li id="n-Сторінка-для-медіа" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%A1%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0_%D0%B4%D0%BB%D1%8F_%D0%BC%D0%B5%D0%B4%D1%96%D0%B0"><span>Сторінка для медіа</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Інструменти</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%81%D0%B8%D0%BB%D0%B0%D0%BD%D0%BD%D1%8F_%D1%81%D1%8E%D0%B4%D0%B8/%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82" title="Перелік усіх сторінок, які посилаються на цю сторінку [j]" accesskey="j"><span>Посилання сюди</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%27%D1%8F%D0%B7%D0%B0%D0%BD%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82" rel="nofollow" title="Останні зміни на сторінках, на які посилається ця сторінка [k]" accesskey="k"><span>Пов'язані редагування</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8" title="Перелік спеціальних сторінок [q]" accesskey="q"><span>Спеціальні сторінки</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&oldid=43048373" title="Постійне посилання на цю версію цієї сторінки"><span>Постійне посилання</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=info" title="Додаткові відомості про цю сторінку"><span>Інформація про сторінку</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A6%D0%B8%D1%82%D0%B0%D1%82%D0%B0&page=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&id=43048373&wpFormIdentifier=titleform" title="Інформація про те, як цитувати цю сторінку"><span>Цитувати сторінку</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:UrlShortener&url=https%3A%2F%2Fuk.wikipedia.org%2Fwiki%2F%25D0%2591%25D1%2596%25D0%25BD%25D0%25BE%25D0%25BC%25D1%2596%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D0%25B8%25D0%25B9_%25D0%25BA%25D0%25BE%25D0%25B5%25D1%2584%25D1%2596%25D1%2586%25D1%2596%25D1%2594%25D0%25BD%25D1%2582"><span>Отримати вкорочену URL-адресу</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fuk.wikipedia.org%2Fwiki%2F%25D0%2591%25D1%2596%25D0%25BD%25D0%25BE%25D0%25BC%25D1%2596%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D0%25B8%25D0%25B9_%25D0%25BA%25D0%25BE%25D0%25B5%25D1%2584%25D1%2596%25D1%2586%25D1%2596%25D1%2594%25D0%25BD%25D1%2582"><span>Завантажити QR-код</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Друк/експорт</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9+%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82"><span>Створити книгу</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:DownloadAsPdf&page=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&action=show-download-screen"><span>Завантажити як PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82&printable=yes" title="Версія цієї сторінки для друку [p]" accesskey="p"><span>Версія до друку</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">В інших проєктах</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Binomial_coefficients" hreflang="en"><span>Вікісховище</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q209875" title="Посилання на пов’язаний елемент сховища даних [g]" accesskey="g"><span>Елемент Вікіданих</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Іншими мовами</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D9%85%D9%84_%D8%B0%D8%A7%D8%AA_%D8%A7%D9%84%D8%AD%D8%AF%D9%8A%D9%86" title="معامل ذات الحدين — арабська" lang="ar" hreflang="ar" data-title="معامل ذات الحدين" data-language-autonym="العربية" data-language-local-name="арабська" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Binomial_%C9%99msallar" title="Binomial əmsallar — азербайджанська" lang="az" hreflang="az" data-title="Binomial əmsallar" data-language-autonym="Azərbaycanca" data-language-local-name="азербайджанська" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%B5%D0%BD_%D0%BA%D0%BE%D0%B5%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82" title="Биномен коефициент — болгарська" lang="bg" hreflang="bg" data-title="Биномен коефициент" data-language-autonym="Български" data-language-local-name="болгарська" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%AA%E0%A6%A6%E0%A7%80_%E0%A6%B8%E0%A6%B9%E0%A6%97" title="দ্বিপদী সহগ — бенгальська" lang="bn" hreflang="bn" data-title="দ্বিপদী সহগ" data-language-autonym="বাংলা" data-language-local-name="бенгальська" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coeficient_binomial" title="Coeficient binomial — каталонська" lang="ca" hreflang="ca" data-title="Coeficient binomial" data-language-autonym="Català" data-language-local-name="каталонська" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%DA%A9%DB%86%D9%84%DA%A9%DB%95%DB%8C_%D8%AF%D9%88%D9%88_%DA%95%D8%A7%D8%AF%DB%95%DB%8C%DB%8C" title="ھاوکۆلکەی دوو ڕادەیی — центральнокурдська" lang="ckb" hreflang="ckb" data-title="ھاوکۆلکەی دوو ڕادەیی" data-language-autonym="کوردی" data-language-local-name="центральнокурдська" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kombina%C4%8Dn%C3%AD_%C4%8D%C3%ADslo" title="Kombinační číslo — чеська" lang="cs" hreflang="cs" data-title="Kombinační číslo" data-language-autonym="Čeština" data-language-local-name="чеська" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC_%D0%BA%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%87%C4%95%D1%81%D0%B5%D0%BC" title="Бином коэффициенчĕсем — чуваська" lang="cv" hreflang="cv" data-title="Бином коэффициенчĕсем" data-language-autonym="Чӑвашла" data-language-local-name="чуваська" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Binomialkoefficient" title="Binomialkoefficient — данська" lang="da" hreflang="da" data-title="Binomialkoefficient" data-language-autonym="Dansk" data-language-local-name="данська" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Binomialkoeffizient" title="Binomialkoeffizient — німецька" lang="de" hreflang="de" data-title="Binomialkoeffizient" data-language-autonym="Deutsch" data-language-local-name="німецька" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CF%89%CE%BD%CF%85%CE%BC%CE%B9%CE%BA%CF%8C%CF%82_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CE%BB%CE%B5%CF%83%CF%84%CE%AE%CF%82" title="Διωνυμικός συντελεστής — грецька" lang="el" hreflang="el" data-title="Διωνυμικός συντελεστής" data-language-autonym="Ελληνικά" data-language-local-name="грецька" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Binomial_coefficient" title="Binomial coefficient — англійська" lang="en" hreflang="en" data-title="Binomial coefficient" data-language-autonym="English" data-language-local-name="англійська" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Binoma_koeficiento" title="Binoma koeficiento — есперанто" lang="eo" hreflang="eo" data-title="Binoma koeficiento" data-language-autonym="Esperanto" data-language-local-name="есперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coeficiente_binomial" title="Coeficiente binomial — іспанська" lang="es" hreflang="es" data-title="Coeficiente binomial" data-language-autonym="Español" data-language-local-name="іспанська" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koefiziente_binomial" title="Koefiziente binomial — баскська" lang="eu" hreflang="eu" data-title="Koefiziente binomial" data-language-autonym="Euskara" data-language-local-name="баскська" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B6%D8%B1%DB%8C%D8%A8_%D8%AF%D9%88%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="ضریب دوجملهای — перська" lang="fa" hreflang="fa" data-title="ضریب دوجملهای" data-language-autonym="فارسی" data-language-local-name="перська" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Binomikerroin" title="Binomikerroin — фінська" lang="fi" hreflang="fi" data-title="Binomikerroin" data-language-autonym="Suomi" data-language-local-name="фінська" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Coefficient_binomial" title="Coefficient binomial — французька" lang="fr" hreflang="fr" data-title="Coefficient binomial" data-language-autonym="Français" data-language-local-name="французька" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coeficiente_binomial" title="Coeficiente binomial — галісійська" lang="gl" hreflang="gl" data-title="Coeficiente binomial" data-language-autonym="Galego" data-language-local-name="галісійська" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A7%D7%93%D7%9D_%D7%91%D7%99%D7%A0%D7%95%D7%9E%D7%99" title="מקדם בינומי — іврит" lang="he" hreflang="he" data-title="מקדם בינומי" data-language-autonym="עברית" data-language-local-name="іврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AA%E0%A4%A6_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BE%E0%A4%82%E0%A4%95" title="द्विपद गुणांक — гінді" lang="hi" hreflang="hi" data-title="द्विपद गुणांक" data-language-autonym="हिन्दी" data-language-local-name="гінді" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Binomni_koeficijent" title="Binomni koeficijent — хорватська" lang="hr" hreflang="hr" data-title="Binomni koeficijent" data-language-autonym="Hrvatski" data-language-local-name="хорватська" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Binomi%C3%A1lis_egy%C3%BCtthat%C3%B3" title="Binomiális együttható — угорська" lang="hu" hreflang="hu" data-title="Binomiális együttható" data-language-autonym="Magyar" data-language-local-name="угорська" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Binomiala_koeficiento" title="Binomiala koeficiento — ідо" lang="io" hreflang="io" data-title="Binomiala koeficiento" data-language-autonym="Ido" data-language-local-name="ідо" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coefficiente_binomiale" title="Coefficiente binomiale — італійська" lang="it" hreflang="it" data-title="Coefficiente binomiale" data-language-autonym="Italiano" data-language-local-name="італійська" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E4%BF%82%E6%95%B0" title="二項係数 — японська" lang="ja" hreflang="ja" data-title="二項係数" data-language-autonym="日本語" data-language-local-name="японська" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%ED%95%AD_%EA%B3%84%EC%88%98" title="이항 계수 — корейська" lang="ko" hreflang="ko" data-title="이항 계수" data-language-autonym="한국어" data-language-local-name="корейська" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC_%D0%BA%D0%BE%D1%8D%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82%D0%B8" title="Бином коэфициенти — киргизька" lang="ky" hreflang="ky" data-title="Бином коэфициенти" data-language-autonym="Кыргызча" data-language-local-name="киргизька" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Co%C3%ABfficiens_binomialis" title="Coëfficiens binomialis — латинська" lang="la" hreflang="la" data-title="Coëfficiens binomialis" data-language-autonym="Latina" data-language-local-name="латинська" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Binomi%C4%81lkoeficients" title="Binomiālkoeficients — латиська" lang="lv" hreflang="lv" data-title="Binomiālkoeficients" data-language-autonym="Latviešu" data-language-local-name="латиська" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Binomiaalco%C3%ABffici%C3%ABnt" title="Binomiaalcoëfficiënt — нідерландська" lang="nl" hreflang="nl" data-title="Binomiaalcoëfficiënt" data-language-autonym="Nederlands" data-language-local-name="нідерландська" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Binomialkoeffisient" title="Binomialkoeffisient — норвезька (букмол)" lang="nb" hreflang="nb" data-title="Binomialkoeffisient" data-language-autonym="Norsk bokmål" data-language-local-name="норвезька (букмол)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Symbol_Newtona" title="Symbol Newtona — польська" lang="pl" hreflang="pl" data-title="Symbol Newtona" data-language-autonym="Polski" data-language-local-name="польська" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coeficiente_binomial" title="Coeficiente binomial — португальська" lang="pt" hreflang="pt" data-title="Coeficiente binomial" data-language-autonym="Português" data-language-local-name="португальська" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coeficient_binomial" title="Coeficient binomial — румунська" lang="ro" hreflang="ro" data-title="Coeficient binomial" data-language-autonym="Română" data-language-local-name="румунська" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82" title="Биномиальный коэффициент — російська" lang="ru" hreflang="ru" data-title="Биномиальный коэффициент" data-language-autonym="Русский" data-language-local-name="російська" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binomni_koeficijent" title="Binomni koeficijent — сербсько-хорватська" lang="sh" hreflang="sh" data-title="Binomni koeficijent" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="сербсько-хорватська" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kombina%C4%8Dn%C3%A9_%C4%8D%C3%ADslo" title="Kombinačné číslo — словацька" lang="sk" hreflang="sk" data-title="Kombinačné číslo" data-language-autonym="Slovenčina" data-language-local-name="словацька" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Binomski_koeficient" title="Binomski koeficient — словенська" lang="sl" hreflang="sl" data-title="Binomski koeficient" data-language-autonym="Slovenščina" data-language-local-name="словенська" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Koeficient%C3%ABt_e_binomit" title="Koeficientët e binomit — албанська" lang="sq" hreflang="sq" data-title="Koeficientët e binomit" data-language-autonym="Shqip" data-language-local-name="албанська" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%BD%D0%B8_%D0%BA%D0%BE%D0%B5%D1%84%D0%B8%D1%86%D0%B8%D1%98%D0%B5%D0%BD%D1%82" title="Биномни коефицијент — сербська" lang="sr" hreflang="sr" data-title="Биномни коефицијент" data-language-autonym="Српски / srpski" data-language-local-name="сербська" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Binomialkoefficient" title="Binomialkoefficient — шведська" lang="sv" hreflang="sv" data-title="Binomialkoefficient" data-language-autonym="Svenska" data-language-local-name="шведська" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%A3%E0%AE%95%E0%AE%AE%E0%AF%8D" title="ஈருறுப்புக் குணகம் — тамільська" lang="ta" hreflang="ta" data-title="ஈருறுப்புக் குணகம்" data-language-autonym="தமிழ்" data-language-local-name="тамільська" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%BC%8F%E4%BF%82%E6%95%B8" title="二項式係數 — китайська" lang="zh" hreflang="zh" data-title="二項式係數" data-language-autonym="中文" data-language-local-name="китайська" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%BC%8F%E4%BF%82%E6%95%B8" title="二項式係數 — кантонська" lang="yue" hreflang="yue" data-title="二項式係數" data-language-autonym="粵語" data-language-local-name="кантонська" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q209875#sitelinks-wikipedia" title="Редагувати міжмовні посилання" class="wbc-editpage">Редагувати посилання</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Цю сторінку востаннє відредаговано о 09:03, 14 липня 2024.</li> <li id="footer-info-copyright">Текст доступний на умовах ліцензії <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.uk">Creative Commons Attribution-ShareAlike</a>; також можуть діяти додаткові 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