Binomial theorem - Wikipedia

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class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Geometric_explanation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_explanation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Geometric explanation</span> </div> </a> <ul id="toc-Geometric_explanation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Binomial_coefficients" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Binomial_coefficients"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Binomial coefficients</span> </div> </a> <button aria-controls="toc-Binomial_coefficients-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Binomial coefficients subsection</span> </button> <ul id="toc-Binomial_coefficients-sublist" class="vector-toc-list"> <li id="toc-Formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Formulas</span> </div> </a> <ul id="toc-Formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Combinatorial interpretation</span> </div> </a> <ul id="toc-Combinatorial_interpretation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proofs" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Proofs</span> </div> </a> <button aria-controls="toc-Proofs-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Proofs subsection</span> </button> <ul id="toc-Proofs-sublist" class="vector-toc-list"> <li id="toc-Combinatorial_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Combinatorial proof</span> </div> </a> <ul id="toc-Combinatorial_proof-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inductive_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductive_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Inductive proof</span> </div> </a> <ul id="toc-Inductive_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Newton's_generalized_binomial_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newton's_generalized_binomial_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Newton's generalized binomial theorem</span> </div> </a> <ul id="toc-Newton's_generalized_binomial_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Further generalizations</span> </div> </a> <ul id="toc-Further_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multinomial_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multinomial_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Multinomial theorem</span> </div> </a> <ul id="toc-Multinomial_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-binomial_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multi-binomial_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Multi-binomial theorem</span> </div> </a> <ul id="toc-Multi-binomial_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_Leibniz_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_Leibniz_rule"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>General Leibniz rule</span> </div> </a> <ul id="toc-General_Leibniz_rule-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Multiple-angle_identities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple-angle_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Multiple-angle identities</span> </div> </a> <ul id="toc-Multiple-angle_identities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Series_for_e" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Series_for_e"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Series for <i>e</i></span> </div> </a> <ul id="toc-Series_for_e-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Probability</span> </div> </a> <ul id="toc-Probability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_abstract_algebra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_abstract_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>In abstract algebra</span> </div> </a> <ul id="toc-In_abstract_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>In popular culture</span> </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Binomial theorem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 70 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-70" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">70 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Binomiaalstelling" title="Binomiaalstelling – Afrikaans" lang="af" hreflang="af" data-title="Binomiaalstelling" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%B0%D8%A7%D8%AA_%D8%A7%D9%84%D8%AD%D8%AF%D9%8A%D9%86" title="مبرهنة ذات الحدين – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة ذات الحدين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</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%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="দ্বিপদী উপপাদ্য – Bangla" lang="bn" hreflang="bn" data-title="দ্বিপদী উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Teorema_binomial" title="Teorema binomial – Banjar" lang="bjn" hreflang="bjn" data-title="Teorema binomial" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%BE%D0%BC_%D0%9D%D1%8C%D1%8E%D1%82%D0%B0%D0%BD%D0%B0" title="Біном Ньютана – Belarusian" lang="be" hreflang="be" data-title="Біном Ньютана" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D1%8E%D1%82%D0%BE%D0%BD%D0%BE%D0%B2_%D0%B1%D0%B8%D0%BD%D0%BE%D0%BC" title="Нютонов бином – Bulgarian" lang="bg" hreflang="bg" data-title="Нютонов бином" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Binomna_teorema" title="Binomna teorema – Bosnian" lang="bs" hreflang="bs" data-title="Binomna teorema" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Binomi_de_Newton" title="Binomi de Newton – Catalan" lang="ca" hreflang="ca" data-title="Binomi de Newton" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D0%B1%D0%B8%D0%BD%D0%BE%D0%BC%C4%95" title="Ньютон биномĕ – Chuvash" lang="cv" hreflang="cv" data-title="Ньютон биномĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Binomick%C3%A1_v%C4%9Bta" title="Binomická věta – Czech" lang="cs" hreflang="cs" data-title="Binomická věta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Binomischer_Lehrsatz" title="Binomischer Lehrsatz – German" lang="de" hreflang="de" data-title="Binomischer Lehrsatz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Newtoni_binoomvalem" title="Newtoni binoomvalem – Estonian" lang="et" hreflang="et" data-title="Newtoni binoomvalem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Διωνυμικό θεώρημα – Greek" lang="el" hreflang="el" data-title="Διωνυμικό θεώρημα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_del_binomio" title="Teorema del binomio – Spanish" lang="es" hreflang="es" data-title="Teorema del binomio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Binomo_de_Newton" title="Binomo de Newton – Esperanto" lang="eo" hreflang="eo" data-title="Binomo de Newton" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Newtonen_binomio" title="Newtonen binomio – Basque" lang="eu" hreflang="eu" data-title="Newtonen binomio" data-language-autonym="Euskara" data-language-local-name="Basque" 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%A8%D8%B3%D8%B7_%D8%AF%D9%88%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="بسط دوجمله‌ای – Persian" lang="fa" hreflang="fa" data-title="بسط دوجمله‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Formule_du_bin%C3%B4me_de_Newton" title="Formule du binôme de Newton – French" lang="fr" hreflang="fr" data-title="Formule du binôme de Newton" data-language-autonym="Français" data-language-local-name="French" 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/Teorema_do_binomio" title="Teorema do binomio – Galician" lang="gl" hreflang="gl" data-title="Teorema do binomio" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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_%EC%A0%95%EB%A6%AC" title="이항 정리 – Korean" lang="ko" hreflang="ko" data-title="이항 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B5%D5%B8%D6%82%D5%BF%D5%B8%D5%B6%D5%AB_%D5%A5%D6%80%D5%AF%D5%A1%D5%B6%D5%A4%D5%A1%D5%B4" title="Նյուտոնի երկանդամ – Armenian" lang="hy" hreflang="hy" data-title="Նյուտոնի երկանդամ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="द्विपद प्रमेय – Hindi" lang="hi" hreflang="hi" data-title="द्विपद प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" 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_pou%C4%8Dak" title="Binomni poučak – Croatian" lang="hr" hreflang="hr" data-title="Binomni poučak" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_binomial" title="Teorema binomial – Indonesian" lang="id" hreflang="id" data-title="Teorema binomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tv%C3%ADli%C3%B0uregla" title="Tvíliðuregla – Icelandic" lang="is" hreflang="is" data-title="Tvíliðuregla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_binomiale" title="Teorema binomiale – Italian" lang="it" hreflang="it" data-title="Teorema binomiale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%91%D7%99%D7%A0%D7%95%D7%9D_%D7%A9%D7%9C_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F" title="הבינום של ניוטון – Hebrew" lang="he" hreflang="he" data-title="הבינום של ניוטון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B3%8D%E0%B2%B5%E0%B2%BF%E0%B2%AA%E0%B2%A6_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B3%87%E0%B2%AF" title="ದ್ವಿಪದ ಪ್ರಮೇಯ – Kannada" lang="kn" hreflang="kn" data-title="ದ್ವಿಪದ ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D0%B1%D0%B8%D0%BD%D0%BE%D0%BC%D1%8B" title="Ньютон биномы – Kazakh" lang="kk" hreflang="kk" data-title="Ньютон биномы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theorema_binomiale" title="Theorema binomiale – Latin" lang="la" hreflang="la" data-title="Theorema binomiale" data-language-autonym="Latina" data-language-local-name="Latin" 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/%C5%85%C5%ABtona_binoms" title="Ņūtona binoms – Latvian" lang="lv" hreflang="lv" data-title="Ņūtona binoms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Binomo_formul%C4%97" title="Binomo formulė – Lithuanian" lang="lt" hreflang="lt" data-title="Binomo formulė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Binomi%C3%A1lis_t%C3%A9tel" title="Binomiális tétel – Hungarian" lang="hu" hreflang="hu" data-title="Binomiális tétel" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Биномна теорема – Macedonian" lang="mk" hreflang="mk" data-title="Биномна теорема" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B5%8D%E0%B4%B5%E0%B4%BF%E0%B4%AA%E0%B4%A6%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B5%87%E0%B4%AF%E0%B4%82" title="ദ്വിപദപ്രമേയം – Malayalam" lang="ml" hreflang="ml" data-title="ദ്വിപദപ്രമേയം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teorem_binomial" title="Teorem binomial – Malay" lang="ms" hreflang="ms" data-title="Teorem binomial" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Binomium_van_Newton" title="Binomium van Newton – Dutch" lang="nl" hreflang="nl" data-title="Binomium van Newton" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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%E5%AE%9A%E7%90%86" title="二項定理 – Japanese" lang="ja" hreflang="ja" data-title="二項定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Binomisk_Formeln" title="Binomisk Formeln – Northern Frisian" lang="frr" hreflang="frr" data-title="Binomisk Formeln" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Binomialformel" title="Binomialformel – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Binomialformel" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Binomialformel" title="Binomialformel – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Binomialformel" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Nyuton_binomi" title="Nyuton binomi – Uzbek" lang="uz" hreflang="uz" data-title="Nyuton binomi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%91%E1%9F%92%E1%9E%9A%E1%9E%B9%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%B8%E1%9E%94%E1%9E%91%E1%9E%91%E1%9F%92%E1%9E%9C%E1%9F%81%E1%9E%92%E1%9E%B6" title="ទ្រឹស្តីបទទ្វេធា – Khmer" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទទ្វេធា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/F%C3%B3rmola_d%C3%ABl_bin%C3%B2mi_%C3%ABd_Newton" title="Fórmola dël binòmi ëd Newton – Piedmontese" lang="pms" hreflang="pms" data-title="Fórmola dël binòmi ëd Newton" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dwumian_Newtona" title="Dwumian Newtona – Polish" lang="pl" hreflang="pl" data-title="Dwumian Newtona" data-language-autonym="Polski" data-language-local-name="Polish" 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/Bin%C3%B3mio_de_Newton" title="Binómio de Newton – Portuguese" lang="pt" hreflang="pt" data-title="Binómio de Newton" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Binomul_lui_Newton" title="Binomul lui Newton – Romanian" lang="ro" hreflang="ro" data-title="Binomul lui Newton" data-language-autonym="Română" data-language-local-name="Romanian" 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%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0" title="Бином Ньютона – Russian" lang="ru" hreflang="ru" data-title="Бином Ньютона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Katrori_i_Binomit" title="Katrori i Binomit – Albanian" lang="sq" hreflang="sq" data-title="Katrori i Binomit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AF%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%B4%E0%B6%AF_%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%9A%E0%B6%BA%E0%B6%BA" title="ද්විපද ප්‍රමේයය – Sinhala" lang="si" hreflang="si" data-title="ද්විපද ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binomial_expansion" title="Binomial expansion – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binomial expansion" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Binomick%C3%A1_veta" title="Binomická veta – Slovak" lang="sk" hreflang="sk" data-title="Binomická veta" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%B1%D8%A7%D9%88%DB%95%DB%8C_%D8%AF%D9%88%D9%88_%D8%AA%DB%8E%D8%B1%D9%85%DB%8C" title="کراوەی دوو تێرمی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کراوەی دوو تێرمی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Биномна теорема – Serbian" lang="sr" hreflang="sr" data-title="Биномна теорема" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binomna_teorema" title="Binomna teorema – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Binomna teorema" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Binomilause" title="Binomilause – Finnish" lang="fi" hreflang="fi" data-title="Binomilause" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Binomialsatsen" title="Binomialsatsen – Swedish" lang="sv" hreflang="sv" data-title="Binomialsatsen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teoremang_binomial" title="Teoremang binomial – Tagalog" lang="tl" hreflang="tl" data-title="Teoremang binomial" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</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%A4%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="ஈருறுப்புத் தேற்றம் – Tamil" lang="ta" hreflang="ta" data-title="ஈருறுப்புத் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Asnful_n_Nyu%E1%B9%ADun" title="Asnful n Nyuṭun – Tachelhit" lang="shi" hreflang="shi" data-title="Asnful n Nyuṭun" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%97%E0%B8%A7%E0%B8%B4%E0%B8%99%E0%B8%B2%E0%B8%A1" title="ทฤษฎีบททวินาม – Thai" lang="th" hreflang="th" data-title="ทฤษฎีบททวินาม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%B8_%D0%A5%D0%B0%D0%B9%D1%91%D0%BC-%D0%9D%D1%8E%D1%82%D0%BE%D0%BD" title="Биноми Хайём-Нютон – Tajik" lang="tg" hreflang="tg" data-title="Биноми Хайём-Нютон" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Binom_a%C3%A7%C4%B1l%C4%B1m%C4%B1" title="Binom açılımı – Turkish" lang="tr" hreflang="tr" data-title="Binom açılımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/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 – Ukrainian" lang="uk" hreflang="uk" data-title="Біном Ньютона" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D9%88_%D8%B1%D9%82%D9%85%DB%8C_%D9%85%D8%B3%D8%A6%D9%84%DB%81_%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA%DB%8C" title="دو رقمی مسئلہ اثباتی – Urdu" lang="ur" hreflang="ur" data-title="دو رقمی مسئلہ اثباتی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_nh%E1%BB%8B_th%E1%BB%A9c" title="Định lý nhị thức – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý nhị thức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%BC%8F%E5%AE%9A%E7%90%86" title="二項式定理 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="二項式定理" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86" title="二项式定理 – Wu" lang="wuu" hreflang="wuu" data-title="二项式定理" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</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%E5%AE%9A%E7%90%86" title="二項式定理 – Cantonese" lang="yue" hreflang="yue" data-title="二項式定理" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86" title="二项式定理 – Chinese" lang="zh" hreflang="zh" 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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic expansion of powers of a binomial</div> <div class="thumb tright" style=""><div class="thumbinner" style="width:217px"><div class="thumbimage noresize" style="width:215px;"> <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{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>2</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>3</mn> <mspace width="1em" /> <mn>3</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>4</mn> <mspace width="1em" /> <mn>6</mn> <mspace width="1em" /> <mn>4</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>5</mn> <mspace width="1em" /> <mn>10</mn> <mspace width="1em" /> <mn>10</mn> <mspace width="1em" /> <mn>5</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>6</mn> <mspace width="1em" /> <mn>15</mn> <mspace width="1em" /> <mn>20</mn> <mspace width="1em" /> <mn>15</mn> <mspace width="1em" /> <mn>6</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> <mn>7</mn> <mspace width="1em" /> <mn>21</mn> <mspace width="1em" /> <mn>35</mn> <mspace width="1em" /> <mn>35</mn> <mspace width="1em" /> <mn>21</mn> <mspace width="1em" /> <mn>7</mn> <mspace width="1em" /> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8beb14cd64d7451f9f9e4f965713d3e7e62cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:30.959ex; height:25.509ex;" alt="{\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}"></span></div><div class="thumbcaption">The <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</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 {\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> appears as the <span class="texhtml mvar" style="font-style:italic;">k</span>th entry in the <span class="texhtml mvar" style="font-style:italic;">n</span>th row of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> (where the top is the 0th row <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}{0}}}"> <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> <mn>0</mn> </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 {0}{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ada13d114ea4fb89f70d0321939cbdc3d864fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.952ex; height:3.509ex;" alt="{\displaystyle {\tbinom {0}{0}}}"></span>). Each entry is the sum of the two above it.</div></div></div> <p>In <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>, the <b>binomial theorem</b> (or <b>binomial expansion</b>) describes the algebraic expansion of <a href="/wiki/Exponentiation" title="Exponentiation">powers</a> of a <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a>. According to the theorem, it is possible to expand the polynomial <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> into a <a href="/wiki/Summation" title="Summation">sum</a> involving terms of the form <span class="texhtml"><i>ax</i><sup><i>b</i></sup><i>y</i><sup><i>c</i></sup></span>, where the exponents <span class="texhtml mvar" style="font-style:italic;">b</span> and <span class="texhtml mvar" style="font-style:italic;">c</span> are <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a> with <span class="texhtml"><i>b</i> + <i>c</i> = <i>n</i></span>, and the <a href="/wiki/Coefficient" title="Coefficient">coefficient</a> <span class="texhtml mvar" style="font-style:italic;">a</span> of each term is a specific <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> depending on <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>. For example, for <span class="texhtml"><i>n</i> = 4</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd0ade2ebe2cf6c1b2a7084b3b5cbf0b8e3a77b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.064ex; height:3.176ex;" alt="{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}"></span> </p><p>The coefficient <span class="texhtml mvar" style="font-style:italic;">a</span> in the term of <span class="texhtml"><i>ax</i><sup><i>b</i></sup><i>y</i><sup><i>c</i></sup></span> is known as the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</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 {\tbinom {n}{b}}}"> <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>b</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}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" 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}{b}}}"></span> or <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}{c}}}"> <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>c</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}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0325ad33b80d9aee734b6a82defe6e5971ec5763" 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}{c}}}"></span> (the two have the same value). These coefficients for varying <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> can be arranged to form <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. These numbers also occur in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, where <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}{b}}}"> <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>b</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}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" 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}{b}}}"></span> gives the number of different <a href="/wiki/Combinations" class="mw-redirect" title="Combinations">combinations</a> (i.e. subsets) of <span class="texhtml mvar" style="font-style:italic;">b</span> <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> that can be chosen from an <span class="texhtml mvar" style="font-style:italic;">n</span>-element <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>. Therefore <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}{b}}}"> <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>b</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}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" 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}{b}}}"></span> is usually pronounced as "<span class="texhtml mvar" style="font-style:italic;">n</span> choose <span class="texhtml mvar" style="font-style:italic;">b</span>". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Special cases of the binomial theorem were known since at least the 4th century BC when <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematician</a> <a href="/wiki/Euclid" title="Euclid">Euclid</a> mentioned the special case of the binomial theorem for exponent <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=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span>.<sup id="cite_ref-Coolidge_1-0" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Greek mathematician <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> cubed various binomials, including <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-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1a88d34243b98b57c4df9db5724f61b59a4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x-1}"></span>.<sup id="cite_ref-Coolidge_1-1" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Indian mathematician <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>'s method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent <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=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"></span>.<sup id="cite_ref-Coolidge_1-2" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting <span class="texhtml mvar" style="font-style:italic;">k</span> objects out of <span class="texhtml mvar" style="font-style:italic;">n</span> without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the <i><a href="/wiki/Chanda%E1%B8%A5%C5%9B%C4%81stra" class="mw-redirect" title="Chandaḥśāstra">Chandaḥśāstra</a></i> by the Indian lyricist <a href="/wiki/Pingala" title="Pingala">Pingala</a> (c. 200 BC), which contains a method for its solution.<sup id="cite_ref-Chinese_2-0" class="reference"><a href="#cite_note-Chinese-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 230">: 230 </span></sup> The commentator <a href="/wiki/Halayudha" title="Halayudha">Halayudha</a> from the 10th century AD explains this method.<sup id="cite_ref-Chinese_2-1" class="reference"><a href="#cite_note-Chinese-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 230">: 230 </span></sup> By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient <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="{\textstyle {\frac {n!}{(n-k)!k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {n!}{(n-k)!k!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca561e86d72abf55deddacd888647f1a6ff751c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.008ex; height:4.343ex;" alt="{\textstyle {\frac {n!}{(n-k)!k!}}}"></span>,<sup id="cite_ref-Biggs_3-0" class="reference"><a href="#cite_note-Biggs-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and a clear statement of this rule can be found in the 12th century text <i>Lilavati</i> by <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhaskara</a>.<sup id="cite_ref-Biggs_3-1" class="reference"><a href="#cite_note-Biggs-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The first known formulation of the binomial theorem and the table of binomial coefficients appears in a work by <a href="/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a>, quoted by <a href="/wiki/Al-Samaw%27al" class="mw-redirect" title="Al-Samaw'al">Al-Samaw'al</a> in his "al-Bahir".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a> described the triangular pattern of the binomial coefficients<sup id="cite_ref-Karaji_7-0" class="reference"><a href="#cite_note-Karaji-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and also provided a <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proof</a> of both the binomial theorem and Pascal's triangle, using an early form of <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<sup id="cite_ref-Karaji_7-1" class="reference"><a href="#cite_note-Karaji-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The Persian poet and mathematician <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> was probably familiar with the formula to higher orders, although many of his mathematical works are lost.<sup id="cite_ref-Coolidge_1-3" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The binomial expansions of small degrees were known in the 13th century mathematical works of <a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and also <a href="/wiki/Chu_Shih-Chieh" class="mw-redirect" title="Chu Shih-Chieh">Chu Shih-Chieh</a>.<sup id="cite_ref-Coolidge_1-4" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Yang Hui attributes the method to a much earlier 11th century text of <a href="/wiki/Jia_Xian" title="Jia Xian">Jia Xian</a>, although those writings are now also lost.<sup id="cite_ref-Chinese_2-2" class="reference"><a href="#cite_note-Chinese-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 142">: 142 </span></sup> </p><p>In 1544, <a href="/wiki/Michael_Stifel" title="Michael Stifel">Michael Stifel</a> introduced the term "binomial coefficient" and showed how to use them to express <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> in terms of <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-1}}"> <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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e01c4a57fbfa8df2bae096bd557fc0d6bb10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.461ex; height:3.176ex;" alt="{\displaystyle (1+x)^{n-1}}"></span>, via "Pascal's triangle".<sup id="cite_ref-Kline_9-0" class="reference"><a href="#cite_note-Kline-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> studied the eponymous triangle comprehensively in his <i>Traité du triangle arithmétique</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a>, and <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a>.<sup id="cite_ref-Kline_9-1" class="reference"><a href="#cite_note-Kline-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>By the early 17th century, some specific cases of the generalized binomial theorem, such as for <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={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30a2a9611deb7d634c6d06f4c40b2c412b442d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.492ex; height:5.176ex;" alt="{\displaystyle n={\frac {1}{2}}}"></span>, can be found in the work of <a href="/wiki/Henry_Briggs_(mathematician)" title="Henry Briggs (mathematician)">Henry Briggs</a>' <i>Arithmetica Logarithmica</i> (1624).<sup id="cite_ref-:0_11-0" class="reference"><a href="#cite_note-:0-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>'s <i>Arithmetic Infinitorum</i> and his method of interpolation.<sup id="cite_ref-Kline_9-2" class="reference"><a href="#cite_note-Kline-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Coolidge_1-5" class="reference"><a href="#cite_note-Coolidge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_11-1" class="reference"><a href="#cite_note-:0-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> A logarithmic version of the theorem for fractional exponents was discovered independently by <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a> who wrote down his formula in 1670.<sup id="cite_ref-:0_11-2" class="reference"><a href="#cite_note-:0-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=2" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to the theorem, the expansion of any nonnegative integer power <span class="texhtml mvar" style="font-style:italic;">n</span> of the binomial <span class="texhtml"><i>x</i> + <i>y</i></span> is a sum of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</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"> <mi>n</mi> <mn>1</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"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</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"> <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"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>y</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> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </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>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</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"> <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"> <mn>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729b7b8a083fff37cab47d7ec3fea0428bc875ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.222ex; height:6.176ex;" alt="{\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},}"></span> where each <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> is a positive integer known as a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 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>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <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>⋯</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 stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f7f454d7e97095674eecb3e641c10fb4eba102" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.044ex; height:6.509ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1}}.}"></span> </p><p>This formula is also referred to as the <b>binomial formula</b> or the <b>binomial identity</b>. Using <a href="/wiki/Capital-sigma_notation" class="mw-redirect" title="Capital-sigma notation">summation notation</a>, it can be written more concisely as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <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>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689f5d27fa52a8ff3f89bed50e7d6625d5c35aaa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.178ex; height:7.009ex;" alt="{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}"></span> </p><p>The final expression follows from the previous one by the symmetry of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, <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="{\textstyle {\binom {n}{k}}={\binom {n}{n-k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <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> <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"> <mi>n</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\binom {n}{k}}={\binom {n}{n-k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769e849eb0d1bbb311309b62f0947c21f4e66f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.112ex; height:3.343ex;" alt="{\textstyle {\binom {n}{k}}={\binom {n}{n-k}}.}"></span> </p><p>A simple variant of the binomial formula is obtained by <a href="/wiki/Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">substituting</a> <span class="texhtml">1</span> for <span class="texhtml mvar" style="font-style:italic;">y</span>, so that it involves only a single <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>. In this form, the formula reads <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+1)^{n}&={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n}\\[4mu]&=\sum _{k=0}^{n}{n \choose k}x^{k}.{\vphantom {\Bigg )}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.522em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></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>0</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>0</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"> <mi>n</mi> <mn>1</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>1</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"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </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> <mo>−</mo> <mn>1</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"> <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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">)</mo> </mrow> </mphantom> </mpadded> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+1)^{n}&={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n}\\[4mu]&=\sum _{k=0}^{n}{n \choose k}x^{k}.{\vphantom {\Bigg )}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c239d57a82431ebf066d336c1996f98bc86d7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:71.566ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}(x+1)^{n}&={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n}\\[4mu]&=\sum _{k=0}^{n}{n \choose k}x^{k}.{\vphantom {\Bigg )}}\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here are the first few cases of the binomial theorem: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+y)^{0}&=1,\\[8pt](x+y)^{1}&=x+y,\\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 1.1em 1.1em 1.1em 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>15</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>21</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>35</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>35</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>21</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>28</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>56</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>70</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>56</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>28</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+y)^{0}&=1,\\[8pt](x+y)^{1}&=x+y,\\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88704e95bcdc63d9e4aa97ef94e2bd7bdf2f28fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.838ex; width:84.034ex; height:44.843ex;" alt="{\displaystyle {\begin{aligned}(x+y)^{0}&=1,\\[8pt](x+y)^{1}&=x+y,\\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.\end{aligned}}}"></span> In general, for the expansion of <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> on the right side in the <span class="texhtml mvar" style="font-style:italic;">n</span>th row (numbered so that the top row is the 0th row): </p> <ul><li>the exponents of <span class="texhtml mvar" style="font-style:italic;">x</span> in the terms are <span class="texhtml"><i>n</i>, <i>n</i> − 1, ..., 2, 1, 0</span> (the last term implicitly contains <span class="texhtml"><i>x</i><sup>0</sup> = 1</span>);</li> <li>the exponents of <span class="texhtml mvar" style="font-style:italic;">y</span> in the terms are <span class="texhtml">0, 1, 2, ..., <i>n</i> − 1, <i>n</i></span> (the first term implicitly contains <span class="texhtml"><i>y</i><sup>0</sup> = 1</span>);</li> <li>the coefficients form the <span class="texhtml mvar" style="font-style:italic;">n</span>th row of Pascal's triangle;</li> <li>before combining like terms, there are <span class="texhtml">2<sup><i>n</i></sup></span> terms <span class="texhtml"><i>x</i><sup><i>i</i></sup><i>y</i><sup><i>j</i></sup></span> in the expansion (not shown);</li> <li>after combining like terms, there are <span class="texhtml"><i>n</i> + 1</span> terms, and their coefficients sum to <span class="texhtml">2<sup><i>n</i></sup></span>.</li></ul> <p>An example illustrating the last two points: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{\text{ terms}})\\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{\text{ terms}})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>x</mi> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mi>y</mi> <mo>+</mo> <mi>y</mi> <mi>x</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>y</mi> <mi>y</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>y</mi> <mi>y</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> terms</mtext> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> terms</mtext> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{\text{ terms}})\\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{\text{ terms}})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeadb038abda179ae54601717c0a878b12dc562e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.645ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{\text{ terms}})\\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{\text{ terms}})\end{aligned}}}"></span> with <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+3+3+1=2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+3+3+1=2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92bcf16be6af24d5c7efb2862702e1586564b049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.486ex; height:2.843ex;" alt="{\displaystyle 1+3+3+1=2^{3}}"></span>. </p><p>A simple example with a specific positive value of <span class="texhtml"><i>y</i></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mo>+</mo> <mn>8.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3173333204fc18788da85394714cc3235b007c87" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.204ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}}"></span> </p><p>A simple example with a specific negative value of <span class="texhtml"><i>y</i></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\\&=x^{3}-6x^{2}+12x-8.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mo>−</mo> <mn>8.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\\&=x^{3}-6x^{2}+12x-8.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ee08a838ea76f9d3f80015bef0e5672dc173dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.204ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\\&=x^{3}-6x^{2}+12x-8.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_explanation">Geometric explanation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=4" title="Edit section: Geometric explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Binomial_theorem_visualisation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/300px-Binomial_theorem_visualisation.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/450px-Binomial_theorem_visualisation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Binomial_theorem_visualisation.svg/600px-Binomial_theorem_visualisation.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Visualisation of binomial expansion up to the 4th power</figcaption></figure> <p>For positive values of <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>, the binomial theorem with <span class="texhtml"><i>n</i> = 2</span> is the geometrically evident fact that a square of side <span class="texhtml"><i>a</i> + <i>b</i></span> can be cut into a square of side <span class="texhtml mvar" style="font-style:italic;">a</span>, a square of side <span class="texhtml mvar" style="font-style:italic;">b</span>, and two rectangles with sides <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>. With <span class="texhtml"><i>n</i> = 3</span>, the theorem states that a cube of side <span class="texhtml"><i>a</i> + <i>b</i></span> can be cut into a cube of side <span class="texhtml mvar" style="font-style:italic;">a</span>, a cube of side <span class="texhtml mvar" style="font-style:italic;">b</span>, three <span class="texhtml"><i>a</i> × <i>a</i> × <i>b</i></span> rectangular boxes, and three <span class="texhtml"><i>a</i> × <i>b</i> × <i>b</i></span> rectangular boxes. </p><p>In <a href="/wiki/Calculus" title="Calculus">calculus</a>, this picture also gives a geometric proof of the <a href="/wiki/Derivative" title="Derivative">derivative</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 (x^{n})'=nx^{n-1}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>n</mi> <msup> <mi>x</mi> <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 (x^{n})'=nx^{n-1}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a8f3f0e543df6b9c279cd71a5b8ad7410ac56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.476ex; height:3.176ex;" alt="{\displaystyle (x^{n})'=nx^{n-1}:}"></span><sup id="cite_ref-barth2004_14-0" class="reference"><a href="#cite_note-barth2004-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> if one sets <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=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60c2905b9231255762355e8d012c24310eeeb667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle a=x}"></span> and <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 b=\Delta x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=\Delta x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12cb218d2dee5518535cfe9ad76e9ce8c4b525ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.008ex; height:2.509ex;" alt="{\displaystyle b=\Delta x,}"></span> interpreting <span class="texhtml mvar" style="font-style:italic;">b</span> as an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> change in <span class="texhtml mvar" style="font-style:italic;">a</span>, then this picture shows the infinitesimal change in the volume of an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional <a href="/wiki/Hypercube" title="Hypercube">hypercube</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 (x+\Delta x)^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <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 (x+\Delta x)^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa338579a15f96549ebfc3df32f1776ee47ac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.11ex; height:2.843ex;" alt="{\displaystyle (x+\Delta x)^{n},}"></span> where the coefficient of the linear term (in <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 \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span>) is <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 nx^{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mi>x</mi> <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 nx^{n-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b98abcaa8edfada62b17114263aa4b05dee16ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.69ex; height:3.009ex;" alt="{\displaystyle nx^{n-1},}"></span> the area of the <span class="texhtml mvar" style="font-style:italic;">n</span> faces, each of dimension <span class="texhtml"><i>n</i> − 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>n</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <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"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eef79a7ec5cd2972686134727454a879c0e5ec2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.291ex; height:6.176ex;" alt="{\displaystyle (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .}"></span> Substituting this into the <a href="/wiki/Definition_of_the_derivative" class="mw-redirect" title="Definition of the derivative">definition of the derivative</a> via a <a href="/wiki/Difference_quotient" title="Difference quotient">difference quotient</a> and taking limits means that the higher order terms, <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 (\Delta x)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Delta x)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e12a3ace37ca8efa855ddefcfdd0e7c81aba52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:3.176ex;" alt="{\displaystyle (\Delta x)^{2}}"></span> and higher, become negligible, and yields the formula <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^{n})'=nx^{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>n</mi> <msup> <mi>x</mi> <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 (x^{n})'=nx^{n-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74a67faa009d02fca64ca3dbe89dc37a20ba27c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.831ex; height:3.176ex;" alt="{\displaystyle (x^{n})'=nx^{n-1},}"></span> interpreted as </p> <dl><dd>"the infinitesimal rate of change in volume of an <span class="texhtml mvar" style="font-style:italic;">n</span>-cube as side length varies is the area of <span class="texhtml mvar" style="font-style:italic;">n</span> of its <span class="texhtml">(<i>n</i> − 1)</span>-dimensional faces".</dd></dl> <p>If one integrates this picture, which corresponds to applying the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, one obtains <a href="/wiki/Cavalieri%27s_quadrature_formula" title="Cavalieri's quadrature formula">Cavalieri's quadrature formula</a>, the integral <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 {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed01a4fee30dbefd2438bcb39a38b52bf8e1a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.855ex; height:3.343ex;" alt="{\displaystyle \textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}}"></span> – see <a href="/wiki/Cavalieri%27s_quadrature_formula#Proof" title="Cavalieri's quadrature formula">proof of Cavalieri's quadrature formula</a> for details.<sup id="cite_ref-barth2004_14-1" class="reference"><a href="#cite_note-barth2004-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Binomial_coefficients">Binomial coefficients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=5" title="Edit section: Binomial coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">Binomial coefficient</a></div> <p>The coefficients that appear in the binomial expansion are called <b>binomial coefficients</b>. These are usually written <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> <mo>,</mo> </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/5eae6e67bc60be3e863cfb51285b0c53d0b93725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.763ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}},}"></span> and pronounced "<span class="texhtml mvar" style="font-style:italic;">n</span> choose <span class="texhtml mvar" style="font-style:italic;">k</span>". </p> <div class="mw-heading mw-heading3"><h3 id="Formulas">Formulas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=6" title="Edit section: Formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coefficient of <span class="texhtml"><i>x</i><sup><i>n</i>−<i>k</i></sup><i>y</i><sup><i>k</i></sup></span> is given by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {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> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {n!}{k!\;(n-k)!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36259d5dfdc824fef49462a783571b54ab8cdc17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.803ex; height:6.343ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!\;(n-k)!}},}"></span> which is defined in terms of the <a href="/wiki/Factorial" title="Factorial">factorial</a> function <span class="texhtml"><i>n</i>!</span>. Equivalently, this formula can be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}}"> <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 stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</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 stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>ℓ</mi> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>ℓ</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>ℓ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b53b7066cde186276197448b9c7384e18dca34a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.103ex; height:7.509ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}}"></span> with <span class="texhtml mvar" style="font-style:italic;">k</span> factors in both the numerator and denominator of the <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a>. Although this formula involves a fraction, the binomial coefficient <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> is actually an <a href="/wiki/Integer" title="Integer">integer</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_interpretation">Combinatorial interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=7" title="Edit section: Combinatorial interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The binomial coefficient <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> can be interpreted as the number of ways to choose <span class="texhtml mvar" style="font-style:italic;">k</span> elements from an <span class="texhtml mvar" style="font-style:italic;">n</span>-element set. This is related to binomials for the following reason: if we write <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> as a <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)(x+y)(x+y)\cdots (x+y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)(x+y)(x+y)\cdots (x+y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179ef002535e281ed387c53e80a73da885948853" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.684ex; height:2.843ex;" alt="{\displaystyle (x+y)(x+y)(x+y)\cdots (x+y),}"></span> then, according to the <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a>, there will be one term in the expansion for each choice of either <span class="texhtml mvar" style="font-style:italic;">x</span> or <span class="texhtml mvar" style="font-style:italic;">y</span> from each of the binomials of the product. For example, there will only be one term <span class="texhtml"><i>x</i><sup><i>n</i></sup></span>, corresponding to choosing <span class="texhtml mvar" style="font-style:italic;">x</span> from each binomial. However, there will be several terms of the form <span class="texhtml"><i>x</i><sup><i>n</i>−2</sup><i>y</i><sup>2</sup></span>, one for each way of choosing exactly two binomials to contribute a <span class="texhtml mvar" style="font-style:italic;">y</span>. Therefore, after <a href="/wiki/Combining_like_terms" class="mw-redirect" title="Combining like terms">combining like terms</a>, the coefficient of <span class="texhtml"><i>x</i><sup><i>n</i>−2</sup><i>y</i><sup>2</sup></span> will be equal to the number of ways to choose exactly <span class="texhtml">2</span> elements from an <span class="texhtml mvar" style="font-style:italic;">n</span>-element set. </p> <div class="mw-heading mw-heading2"><h2 id="Proofs">Proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=8" title="Edit section: Proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_proof">Combinatorial proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=9" title="Edit section: Combinatorial proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Expanding <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> yields the sum of the <span class="texhtml">2<sup><i>n</i></sup></span> products of the form <span class="texhtml"><i>e</i><sub>1</sub><i>e</i><sub>2</sub> ... <i>e</i><sub><i>n</i></sub></span> where each <span class="texhtml"><i>e</i><sub><i>i</i></sub></span> is <span class="texhtml mvar" style="font-style:italic;"><i>x</i></span> or <span class="texhtml mvar" style="font-style:italic;">y</span>. Rearranging factors shows that each product equals <span class="texhtml"><i>x</i><sup><i>n</i>−<i>k</i></sup><i>y</i><sup><i>k</i></sup></span> for some <span class="texhtml mvar" style="font-style:italic;">k</span> between <span class="texhtml">0</span> and <span class="texhtml mvar" style="font-style:italic;">n</span>. For a given <span class="texhtml mvar" style="font-style:italic;">k</span>, the following are proved equal in succession: </p> <ul><li>the number of terms equal to <span class="texhtml"><i>x</i><sup><i>n</i>−<i>k</i></sup><i>y</i><sup><i>k</i></sup></span> in the expansion</li> <li>the number of <span class="texhtml mvar" style="font-style:italic;">n</span>-character <span class="texhtml"><i>x</i>,<i>y</i></span> strings having <span class="texhtml mvar" style="font-style:italic;">y</span> in exactly <span class="texhtml mvar" style="font-style:italic;">k</span> positions</li> <li>the number of <span class="texhtml mvar" style="font-style:italic;">k</span>-element subsets of <span class="texhtml">{1, 2, ..., <i>n</i>}</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 {\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> <mo>,</mo> </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/5eae6e67bc60be3e863cfb51285b0c53d0b93725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.763ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}},}"></span> either by definition, or by a short combinatorial argument if one is defining <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> as <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 {\tfrac {n!}{k!(n-k)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n!}{k!(n-k)!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730cecb69947b761825b5e5dce68ed70c0ddcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.655ex; height:4.343ex;" alt="{\displaystyle {\tfrac {n!}{k!(n-k)!}}.}"></span></li></ul> <p>This proves the binomial theorem. </p> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=10" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coefficient of <span class="texhtml"><i>xy</i><sup>2</sup></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>x</mi> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>x</mi> <mi>y</mi> <mi>y</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo>+</mo> <mi>y</mi> <mi>x</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>y</mi> <mi>x</mi> <mi>y</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>y</mi> <mi>y</mi> <mi>x</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo>+</mo> <mi>y</mi> <mi>y</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>_</mo> </munder> </mrow> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6baba2e07c3abba2bc13af09155fb612b985d4a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.377ex; margin-bottom: -0.794ex; width:61.749ex; height:11.009ex;" alt="{\displaystyle {\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}}"></span> equals <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 {3}{2}}=3}"> <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>3</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {3}{2}}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdb4fdc4b88edb1bf7a95169940b773b896eb07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.213ex; height:3.343ex;" alt="{\displaystyle {\tbinom {3}{2}}=3}"></span> because there are three <span class="texhtml"><i>x</i>,<i>y</i></span> strings of length 3 with exactly two <span class="texhtml mvar" style="font-style:italic;">y</span>'s, namely, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyy,\;yxy,\;yyx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mi>y</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyy,\;yxy,\;yyx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b905eafded25f36a6656fc2f7b09d1768639c00f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.927ex; height:2.009ex;" alt="{\displaystyle xyy,\;yxy,\;yyx,}"></span> corresponding to the three 2-element subsets of <span class="texhtml">{1, 2, 3}</span>, namely, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3\},\;\{1,3\},\;\{1,2\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3\},\;\{1,3\},\;\{1,2\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64513f22d5956e2aa53b1076c744dcc0867441a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.056ex; height:2.843ex;" alt="{\displaystyle \{2,3\},\;\{1,3\},\;\{1,2\},}"></span> where each subset specifies the positions of the <span class="texhtml mvar" style="font-style:italic;">y</span> in a corresponding string. </p> <div class="mw-heading mw-heading3"><h3 id="Inductive_proof">Inductive proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=11" title="Edit section: Inductive proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Mathematical_induction" title="Mathematical induction">Induction</a> yields another proof of the binomial theorem. When <span class="texhtml"><i>n</i> = 0</span>, both sides equal <span class="texhtml">1</span>, since <span class="texhtml"><i>x</i><sup>0</sup> = 1</span> and <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}{0}}=1.}"> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {0}{0}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d5094df91be1f7ac784c31924000326df7ac09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.86ex; height:3.509ex;" alt="{\displaystyle {\tbinom {0}{0}}=1.}"></span> Now suppose that the equality holds for a given <span class="texhtml mvar" style="font-style:italic;">n</span>; we will prove it for <span class="texhtml"><i>n</i> + 1</span>. For <span class="texhtml"><i>j</i>, <i>k</i> ≥ 0</span>, let <span class="texhtml">[<i>f</i>(<i>x</i>, <i>y</i>)]<sub><i>j</i>,<i>k</i></sub></span> denote the coefficient of <span class="texhtml"><i>x</i><sup><i>j</i></sup><i>y</i><sup><i>k</i></sup></span> in the polynomial <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>)</span>. By the inductive hypothesis, <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> is a polynomial in <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> such that <span class="texhtml">[(<i>x</i> + <i>y</i>)<sup><i>n</i></sup>]<sub><i>j</i>,<i>k</i></sub></span> is <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> if <span class="texhtml"><i>j</i> + <i>k</i> = <i>n</i></span>, and <span class="texhtml mvar" style="font-style:italic;">0</span> otherwise. The identity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65e340981c1654d6abc6b8cee9999f33670f160" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.585ex; height:3.176ex;" alt="{\displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}}"></span> shows that <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i>+1</sup></span> is also a polynomial in <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518ebfbee0b81ffcd211d9d6bc1bd574da3e1f40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.499ex; height:3.343ex;" alt="{\displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},}"></span> since if <span class="texhtml"><i>j</i> + <i>k</i> = <i>n</i> + 1</span>, then <span class="texhtml">(<i>j</i> − 1) + <i>k</i> = <i>n</i></span> and <span class="texhtml"><i>j</i> + (<i>k</i> − 1) = <i>n</i></span>. Now, the right hand side is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{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"> <mi>n</mi> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{k}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734f769b5f7a2aebf3f86f4c49507d8671ef26a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.856ex; height:6.176ex;" alt="{\displaystyle {\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{k}},}"></span> by <a href="/wiki/Pascal%27s_identity" class="mw-redirect" title="Pascal's identity">Pascal's identity</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> On the other hand, if <span class="texhtml"><i>j</i> + <i>k</i> ≠ <i>n</i> + 1</span>, then <span class="texhtml">(<i>j</i> – 1) + <i>k</i> ≠ <i>n</i></span> and <span class="texhtml"><i>j</i> + (<i>k</i> – 1) ≠ <i>n</i></span>, so we get <span class="texhtml">0 + 0 = 0</span>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mo>+</mo> <mn>1</mn> </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>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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddddaf5cf3d320362761c7c6647f9a1cf2d380e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.793ex; height:7.509ex;" alt="{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},}"></span> which is the inductive hypothesis with <span class="texhtml"><i>n</i> + 1</span> substituted for <span class="texhtml mvar" style="font-style:italic;">n</span> and so completes the inductive step. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=12" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Newton's_generalized_binomial_theorem"><span id="Newton.27s_generalized_binomial_theorem"></span>Newton's generalized binomial theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=13" title="Edit section: Newton's generalized binomial theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_series" title="Binomial series">Binomial series</a></div> <p>Around 1665, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to <a href="/wiki/Complex_number" title="Complex number">complex</a> exponents.) In this generalization, the finite sum is replaced by an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number <span class="texhtml mvar" style="font-style:italic;">r</span>, one can define <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{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>r</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>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>r</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd683d7380bc8b42be630f3e8076ba5e0782ec0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.414ex; height:6.343ex;" alt="{\displaystyle {r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},}"></span> where <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 (\cdot )_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot )_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a78b2be94cc3feb64ce1412026c2c2731865159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.545ex; height:2.843ex;" alt="{\displaystyle (\cdot )_{k}}"></span> is the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>, here standing for a <a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>. This agrees with the usual definitions when <span class="texhtml mvar" style="font-style:italic;">r</span> is a nonnegative integer. Then, if <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are real numbers with <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| > |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i></span>|</span>,<sup id="cite_ref-convergence_16-0" class="reference"><a href="#cite_note-convergence-16"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> and <span class="texhtml mvar" style="font-style:italic;">r</span> is any complex number, one has <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <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 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>r</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>r</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mi>r</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e6a624eb97d705f85fe3f8697c1d74809043b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.656ex; margin-bottom: -0.182ex; width:73.578ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}}"></span> </p><p>When <span class="texhtml mvar" style="font-style:italic;">r</span> is a nonnegative integer, the binomial coefficients for <span class="texhtml"><i>k</i> > <i>r</i></span> are zero, so this equation reduces to the usual binomial theorem, and there are at most <span class="texhtml"><i>r</i> + 1</span> nonzero terms. For other values of <span class="texhtml mvar" style="font-style:italic;">r</span>, the series typically has infinitely many nonzero terms. </p><p>For example, <span class="texhtml"><i>r</i> = 1/2</span> gives the following series for the square root: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>128</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>256</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4840d862419d0ca7e32fe6a928dcb031a5ee075b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.999ex; height:5.176ex;" alt="{\displaystyle {\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots .}"></span> </p><p>Taking <span class="texhtml"><i>r</i> = −1</span>, the generalized binomial series gives the <a href="/wiki/Geometric_series#Sum" title="Geometric series">geometric series formula</a>, valid for <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| < 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .}"> <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> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ae01ea9d6a90116b7231962e3bc09ea998eae9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.668ex; height:5.343ex;" alt="{\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .}"></span> </p><p>More generally, with <span class="texhtml"><i>r</i> = −<i>s</i></span>, we have for <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| < 1</span>:<sup id="cite_ref-wolfram2_17-0" class="reference"><a href="#cite_note-wolfram2-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1+x)^{s}}}=\sum _{k=0}^{\infty }{-s \choose k}x^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}x^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <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> </mrow> </mfrac> </mrow> <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 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"> <mrow> <mo>−</mo> <mi>s</mi> </mrow> <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> <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"> <mrow> <mi>s</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> <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> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1+x)^{s}}}=\sum _{k=0}^{\infty }{-s \choose k}x^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}x^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fea5f3381e7620bf8bdbddf1d4b254b872a716" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.901ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{(1+x)^{s}}}=\sum _{k=0}^{\infty }{-s \choose k}x^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}x^{k}.}"></span> </p><p>So, for instance, when <span class="texhtml"><i>s</i> = 1/2</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>35</mn> <mn>128</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>63</mn> <mn>256</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecd189e961870d54d4cb08cced5ef3c13d12ec4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.836ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots .}"></span> </p><p>Replacing <span class="texhtml mvar" style="font-style:italic;">x</span> with <span class="texhtml mvar" style="font-style:italic;">-x</span> yields: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}(-x)^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <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> </mrow> </mfrac> </mrow> <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 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"> <mrow> <mi>s</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> <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 stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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 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"> <mrow> <mi>s</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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}(-x)^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093027c78cfc7cfc5700567e872995ad89c3f53e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.764ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}(-x)^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.}"></span> </p><p>So, for instance, when <span class="texhtml"><i>s</i> = 1/2</span>, we have for <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| < 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {1-x}}}=1+{\frac {1}{2}}x+{\frac {3}{8}}x^{2}+{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}+{\frac {63}{256}}x^{5}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mi>x</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>35</mn> <mn>128</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>63</mn> <mn>256</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {1-x}}}=1+{\frac {1}{2}}x+{\frac {3}{8}}x^{2}+{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}+{\frac {63}{256}}x^{5}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c75ee0bd173f2685833880e9554a754c3b2b1ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.836ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {1-x}}}=1+{\frac {1}{2}}x+{\frac {3}{8}}x^{2}+{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}+{\frac {63}{256}}x^{5}+\cdots .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Further_generalizations">Further generalizations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=14" title="Edit section: Further generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generalized binomial theorem can be extended to the case where <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are complex numbers. For this version, one should again assume <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| > |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i></span>|</span><sup id="cite_ref-convergence_16-1" class="reference"><a href="#cite_note-convergence-16"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> and define the powers of <span class="texhtml"><i>x</i> + <i>y</i></span> and <span class="texhtml mvar" style="font-style:italic;">x</span> using a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> <a href="/wiki/Complex_logarithm" title="Complex logarithm">branch of log</a> defined on an open disk of radius <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>|</span> centered at <span class="texhtml mvar" style="font-style:italic;">x</span>. The generalized binomial theorem is valid also for elements <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> of a <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a> as long as <span class="texhtml"><i>xy</i> = <i>yx</i></span>, and <span class="texhtml mvar" style="font-style:italic;">x</span> is invertible, and <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i>/<i>x</i></span>‖ < 1</span>. </p><p>A version of the binomial theorem is valid for the following <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>-like family of polynomials: for a given real constant <span class="texhtml mvar" style="font-style:italic;">c</span>, define <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^{(0)}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{(0)}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/210534920fcae003a0cfd8a7e7074500094630ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.924ex; height:2.843ex;" alt="{\displaystyle x^{(0)}=1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <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> <mo stretchy="false">[</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>c</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fb041571bc1f80d3b23d148bd04e4245ebf4c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.389ex; height:6.843ex;" alt="{\displaystyle x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]}"></span> for <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ab13b00b6f8bb19aea0dcfcc78f18a2b6cbffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n>0.}"></span> Then<sup id="cite_ref-Sokolowsky_18-0" class="reference"><a href="#cite_note-Sokolowsky-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438478762c66617f579457f02b2bf8cb6834be43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.906ex; height:7.009ex;" alt="{\displaystyle (a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.}"></span> The case <span class="texhtml"><i>c</i> = 0</span> recovers the usual binomial theorem. </p><p>More generally, a sequence <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_{n}\}_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p_{n}\}_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54086a60a49b9e526eba4fa14d8fa5387ec570bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.032ex; height:3.009ex;" alt="{\displaystyle \{p_{n}\}_{n=0}^{\infty }}"></span> of polynomials is said to be <b>of binomial type</b> if </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 \deg p_{n}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg p_{n}=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20bba5a77b36a9e089b812bee18420195b25e8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.756ex; height:2.509ex;" alt="{\displaystyle \deg p_{n}=n}"></span> for all <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>,</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 p_{0}(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca86f7c8dc96bc75d6489896ab25f16749541fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.546ex; height:2.843ex;" alt="{\displaystyle p_{0}(0)=1}"></span>, and</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 p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)p_{n-k}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <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> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)p_{n-k}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17fd2f1d1e2db8bffba31b9f2f84c064cd52a7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:34.153ex; height:7.009ex;" alt="{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)p_{n-k}(y)}"></span> for all <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>, <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, and <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>.</li></ul> <p>An operator <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> on the space of polynomials is said to be the <i>basis operator</i> of the sequence <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_{n}\}_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p_{n}\}_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54086a60a49b9e526eba4fa14d8fa5387ec570bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.032ex; height:3.009ex;" alt="{\displaystyle \{p_{n}\}_{n=0}^{\infty }}"></span> if <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 Qp_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Qp_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d571e6906c0665fef827d2ed3f2c6312a2ca79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.323ex; height:2.509ex;" alt="{\displaystyle Qp_{0}=0}"></span> and <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 Qp_{n}=np_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Qp_{n}=np_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8643cb89a572e2658c07c220ee8672f7dc960f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.208ex; height:2.509ex;" alt="{\displaystyle Qp_{n}=np_{n-1}}"></span> for all <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\geqslant 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geqslant 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4988f75f48013d159669b6725b19df177ff8a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geqslant 1}"></span>. A sequence <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_{n}\}_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p_{n}\}_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54086a60a49b9e526eba4fa14d8fa5387ec570bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.032ex; height:3.009ex;" alt="{\displaystyle \{p_{n}\}_{n=0}^{\infty }}"></span> is binomial if and only if its basis operator is a <a href="/wiki/Delta_operator" title="Delta operator">Delta operator</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Writing <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 E^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8f0535fc3d0ec87a96f2f60ddb7e8b54d08837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.896ex; height:2.343ex;" alt="{\displaystyle E^{a}}"></span> for the shift by <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> operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference <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-E^{-c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>−</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I-E^{-c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7ace1f991139eb5914e71c2eacffb639d12dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.029ex; height:2.676ex;" alt="{\displaystyle I-E^{-c}}"></span> for <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}"></span>, the ordinary derivative for <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ee918699d0cb4b8c633cc1f520a8a7a174f44a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=0}"></span>, and the forward difference <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 E^{-c}-I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>c</mi> </mrow> </msup> <mo>−</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{-c}-I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e09244ef0e9061fcaa84b1c6a87cbd856736a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.029ex; height:2.676ex;" alt="{\displaystyle E^{-c}-I}"></span> for <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a48dc798956117afd8c429c39886678c0e7204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c<0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Multinomial_theorem">Multinomial theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=15" title="Edit section: Multinomial theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multinomial_theorem" title="Multinomial theorem">Multinomial theorem</a></div> <p>The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <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> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c8a333c04118aa1700b34a9ccc135ebce0c96d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:70.573ex; height:7.009ex;" alt="{\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},}"></span> </p><p>where the summation is taken over all sequences of nonnegative integer indices <span class="texhtml"><i>k</i><sub>1</sub></span> through <span class="texhtml"><i>k</i><sub><i>m</i></sub></span> such that the sum of all <span class="texhtml"><i>k</i><sub><i>i</i></sub></span> is <span class="texhtml mvar" style="font-style:italic;">n</span>. (For each term in the expansion, the exponents must add up to <span class="texhtml mvar" style="font-style:italic;">n</span>). The coefficients <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_{1},\cdots ,k_{m}}}}"> <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> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </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_{1},\cdots ,k_{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9d48818372d704aece15c194ee5058683e1827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.85ex; height:3.509ex;" alt="{\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}}"></span> are known as multinomial coefficients, and can be computed by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.}"> <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> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </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> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>⋅</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> <mo>⋯</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8321b2ae29fbff7b32d3e42d6f490ef426c62e4e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.166ex; height:6.176ex;" alt="{\displaystyle {\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.}"></span> </p><p>Combinatorially, the multinomial coefficient <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_{1},\cdots ,k_{m}}}}"> <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> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </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_{1},\cdots ,k_{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9d48818372d704aece15c194ee5058683e1827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.85ex; height:3.509ex;" alt="{\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}}"></span> counts the number of different ways to <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> an <span class="texhtml mvar" style="font-style:italic;">n</span>-element set into <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> <a href="/wiki/Subset" title="Subset">subsets</a> of sizes <span class="texhtml"><i>k</i><sub>1</sub>, ..., <i>k</i><sub><i>m</i></sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Multi-binomial_theorem"><span class="anchor" id="multi-binomial"></span> Multi-binomial theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=16" title="Edit section: Multi-binomial theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </munderover> <mo>⋯</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </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"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <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>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc586b38c2bfeda5a54e21c9943b91a3e0ffd5a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:75.617ex; height:7.509ex;" alt="{\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.}"></span> </p><p>This may be written more concisely, by <a href="/wiki/Multi-index_notation" title="Multi-index notation">multi-index notation</a>, as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>≤</mo> <mi>α</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>α</mi> <mi>ν</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>ν</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mi>ν</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f58bf8c6d48ede3217ac8dc0866a486537495" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.843ex; height:6.843ex;" alt="{\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="General_Leibniz_rule">General Leibniz rule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=17" title="Edit section: General Leibniz rule"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz rule</a></div> <p>The general Leibniz rule gives the <span class="texhtml mvar" style="font-style:italic;">n</span>th derivative of a product of two functions in a form similar to that of the binomial theorem:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d595f0b7cea67e8f6be661f00b64253960e90e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.862ex; height:7.009ex;" alt="{\displaystyle (fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).}"></span> </p><p>Here, the superscript <span class="texhtml">(<i>n</i>)</span> indicates the <span class="texhtml mvar" style="font-style:italic;">n</span>th derivative of a function, <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^{(n)}(x)={\tfrac {d^{n}}{dx^{n}}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(n)}(x)={\tfrac {d^{n}}{dx^{n}}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8be2089e38200b1c75c76ed4125ca904b745dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.074ex; height:3.843ex;" alt="{\displaystyle f^{(n)}(x)={\tfrac {d^{n}}{dx^{n}}}f(x)}"></span>. If one sets <span class="texhtml"><i>f</i>(<i>x</i>) = <i>e</i><sup><i>ax</i></sup></span> and <span class="texhtml"><i>g</i>(<i>x</i>) = <i>e</i><sup><i>bx</i></sup></span>, cancelling the common factor of <span class="texhtml"><i>e</i><sup>(<i>a</i> + <i>b</i>)<i>x</i></sup></span> from each term gives the ordinary binomial theorem.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=18" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Multiple-angle_identities">Multiple-angle identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=19" title="Edit section: Multiple-angle identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> the binomial theorem can be combined with <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a> to yield <a href="/wiki/List_of_trigonometric_identities#Multiple-angle_formulae" title="List of trigonometric identities">multiple-angle formulas</a> for the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>. According to De Moivre's formula, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266020d64fdd0a87833d6e1a547aca69f289df13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.267ex; height:3.009ex;" alt="{\displaystyle \cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.}"></span> </p><p>Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for <span class="texhtml">cos(<i>nx</i>)</span> and <span class="texhtml">sin(<i>nx</i>)</span>. For example, since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x=(\cos ^{2}x-\sin ^{2}x)+i(2\cos x\sin x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x=(\cos ^{2}x-\sin ^{2}x)+i(2\cos x\sin x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0463b5ec5e31731293292ed6cead66ff9078acba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.414ex; height:3.343ex;" alt="{\displaystyle \left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x=(\cos ^{2}x-\sin ^{2}x)+i(2\cos x\sin x),}"></span> But De Moivre's formula identifies the left side with <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 (\cos x+i\sin x)^{2}=\cos(2x)+i\sin(2x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos x+i\sin x)^{2}=\cos(2x)+i\sin(2x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5499cc6a353cfe1c9bfb90acf369559f0511a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.992ex; height:3.176ex;" alt="{\displaystyle (\cos x+i\sin x)^{2}=\cos(2x)+i\sin(2x)}"></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e572b690cc6e28c7f4d67c2965107beb7e59fd17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.879ex; height:3.176ex;" alt="{\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,}"></span> which are the usual double-angle identities. Similarly, since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>i</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ca454c2d40578ddeb2ee667545a157705b4510" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.843ex; height:3.343ex;" alt="{\displaystyle \left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,}"></span> De Moivre's formula yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ad61b8f63faff5738968943541229eee97efb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.165ex; height:3.176ex;" alt="{\displaystyle \cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.}"></span> In general, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </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>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0462603fc93a358b91212929e612d326b43c11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.367ex; height:6.676ex;" alt="{\displaystyle \cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </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>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b08921a63a30c7f5e06a6b80c9b21ac87b3957" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.547ex; height:6.676ex;" alt="{\displaystyle \sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.}"></span>There are also similar formulas using <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Series_for_e">Series for <i>e</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=20" title="Edit section: Series for e"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">number <span class="texhtml mvar" style="font-style:italic;">e</span></a> is often defined by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de79f5ee2d494650fd37239dbf6b99864504c93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.362ex; height:6.176ex;" alt="{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}"></span> </p><p>Applying the binomial theorem to this expression yields the usual <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> for <span class="texhtml mvar" style="font-style:italic;">e</span>. In particular: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1+{\frac {1}{n}}\right)^{n}=1+{n \choose 1}{\frac {1}{n}}+{n \choose 2}{\frac {1}{n^{2}}}+{n \choose 3}{\frac {1}{n^{3}}}+\cdots +{n \choose n}{\frac {1}{n^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <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"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </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>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+{\frac {1}{n}}\right)^{n}=1+{n \choose 1}{\frac {1}{n}}+{n \choose 2}{\frac {1}{n^{2}}}+{n \choose 3}{\frac {1}{n^{3}}}+\cdots +{n \choose n}{\frac {1}{n^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f2abc57791e4fa80bd191afba68fd055ed321a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.22ex; height:6.176ex;" alt="{\displaystyle \left(1+{\frac {1}{n}}\right)^{n}=1+{n \choose 1}{\frac {1}{n}}+{n \choose 2}{\frac {1}{n^{2}}}+{n \choose 3}{\frac {1}{n^{3}}}+\cdots +{n \choose n}{\frac {1}{n^{n}}}.}"></span> </p><p>The <span class="texhtml mvar" style="font-style:italic;">k</span>th term of this sum is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}\cdot {\frac {n(n-1)(n-2)\cdots (n-k+1)}{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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <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>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}\cdot {\frac {n(n-1)(n-2)\cdots (n-k+1)}{n^{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1f51da5abc3b37abc9d33aac26f394b08c4f6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.008ex; height:6.343ex;" alt="{\displaystyle {n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}\cdot {\frac {n(n-1)(n-2)\cdots (n-k+1)}{n^{k}}}}"></span> </p><p>As <span class="texhtml"><i>n</i> → ∞</span>, the rational expression on the right approaches <span class="texhtml">1</span>, and therefore <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bf5bae433c544b5e779e6abe04f9ce52481162" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.235ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }{n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}.}"></span> </p><p>This indicates that <span class="texhtml mvar" style="font-style:italic;">e</span> can be written as a series: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661bb44406247d11db4a81161f8a6f4ad3ff4ca1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.417ex; height:7.009ex;" alt="{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots .}"></span> </p><p>Indeed, since each term of the binomial expansion is an <a href="/wiki/Monotonic_function" title="Monotonic function">increasing function</a> of <span class="texhtml mvar" style="font-style:italic;">n</span>, it follows from the <a href="/wiki/Monotone_convergence_theorem" title="Monotone convergence theorem">monotone convergence theorem</a> for series that the sum of this infinite series is equal to <span class="texhtml mvar" style="font-style:italic;">e</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Probability">Probability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=21" title="Edit section: Probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The binomial theorem is closely related to the probability mass function of the <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a>. The probability of a (countable) collection of independent Bernoulli trials <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_{t}\}_{t\in S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\in S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec25a5651e14c9854b326be4e03657bf6a8fac36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.057ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\in S}}"></span> with probability of success <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\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c3a52aa7b2d00227e85c641cca67e85583c43c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.752ex; height:2.843ex;" alt="{\displaystyle p\in [0,1]}"></span> all not happening is </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\left(\bigcap _{t\in S}X_{t}^{C}\right)=(1-p)^{|S|}=\sum _{n=0}^{|S|}{|S| \choose n}(-p)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>p</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 P\left(\bigcap _{t\in S}X_{t}^{C}\right)=(1-p)^{|S|}=\sum _{n=0}^{|S|}{|S| \choose n}(-p)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8118e1c91569e8f922d11a8fb040be0696f0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.423ex; height:7.843ex;" alt="{\displaystyle P\left(\bigcap _{t\in S}X_{t}^{C}\right)=(1-p)^{|S|}=\sum _{n=0}^{|S|}{|S| \choose n}(-p)^{n}.}"></span></dd></dl> <p>An upper bound for this quantity is <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 e^{-p|S|}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-p|S|}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adc7632a9de8ffd65a9266df86306c17182bfe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.843ex;" alt="{\displaystyle e^{-p|S|}.}"></span><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_abstract_algebra">In abstract algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=22" title="Edit section: In abstract algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The binomial theorem is valid more generally for two elements <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> in a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, or even a <a href="/wiki/Semiring" title="Semiring">semiring</a>, provided that <span class="texhtml"><i>xy</i> = <i>yx</i></span>. For example, it holds for two <span class="texhtml"><i>n</i> × <i>n</i></span> matrices, provided that those matrices commute; this is useful in computing powers of a matrix.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>The binomial theorem can be stated by saying that the <a href="/wiki/Polynomial_sequence" title="Polynomial sequence">polynomial sequence</a> <span class="texhtml">{1, <i>x</i>, <i>x</i><sup>2</sup>, <i>x</i><sup>3</sup>, ...}</span> is of <a href="/wiki/Binomial_type" title="Binomial type">binomial type</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=23" title="Edit section: In popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The binomial theorem is mentioned in the <a href="/wiki/Major-General%27s_Song" title="Major-General's Song">Major-General's Song</a> in the comic opera <i><a href="/wiki/The_Pirates_of_Penzance" title="The Pirates of Penzance">The Pirates of Penzance</a></i>.</li> <li><a href="/wiki/Professor_Moriarty" title="Professor Moriarty">Professor Moriarty</a> is described by Sherlock Holmes as having written <a href="/wiki/A_Treatise_on_the_Binomial_Theorem" title="A Treatise on the Binomial Theorem">a treatise on the binomial theorem</a>.</li> <li>The Portuguese poet <a href="/wiki/Fernando_Pessoa" title="Fernando Pessoa">Fernando Pessoa</a>, using the heteronym <a href="/wiki/%C3%81lvaro_de_Campos" title="Álvaro de Campos">Álvaro de Campos</a>, wrote that "Newton's Binomial is as beautiful as the <i><a href="/wiki/Venus_de_Milo" title="Venus de Milo">Venus de Milo</a></i>. The truth is that few people notice it."<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>In the 2014 film <i><a href="/wiki/The_Imitation_Game" title="The Imitation Game">The Imitation Game</a></i>, Alan Turing makes reference to Isaac Newton's work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Binomial_approximation" title="Binomial approximation">Binomial approximation</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Binomial_inverse_theorem" class="mw-redirect" title="Binomial inverse theorem">Binomial inverse theorem</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li> <li><a href="/wiki/Tannery%27s_theorem" title="Tannery's theorem">Tannery's theorem</a></li> <li><a href="/wiki/Polynomials_calculating_sums_of_powers_of_arithmetic_progressions" class="mw-redirect" title="Polynomials calculating sums of powers of arithmetic progressions">Polynomials calculating sums of powers of arithmetic progressions</a></li> <li><a href="/wiki/Gaussian_binomial_coefficient#q-binomial_theorem" title="Gaussian binomial coefficient">q-binomial theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=25" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-convergence-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-convergence_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-convergence_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">This is to guarantee convergence. Depending on <span class="texhtml mvar" style="font-style:italic;">r</span>, the series may also converge sometimes when <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i></span>|</span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Coolidge-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Coolidge_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Coolidge_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Coolidge_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Coolidge_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Coolidge_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Coolidge_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.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-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}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCoolidge1949" class="citation journal cs1">Coolidge, J. L. (1949). "The Story of the Binomial Theorem". <i>The American Mathematical Monthly</i>. <b>56</b> (3): 147–157. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2305028">10.2307/2305028</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2305028">2305028</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Story+of+the+Binomial+Theorem&rft.volume=56&rft.issue=3&rft.pages=147-157&rft.date=1949&rft_id=info%3Adoi%2F10.2307%2F2305028&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2305028%23id-name%3DJSTOR&rft.aulast=Coolidge&rft.aufirst=J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Chinese-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chinese_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chinese_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Chinese_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean-Claude_MartzloffS.S._WilsonJ._GernetJ._Dhombres1987" class="citation book cs1">Jean-Claude Martzloff; S.S. Wilson; J. Gernet; J. Dhombres (1987). <i>A history of Chinese mathematics</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+Chinese+mathematics&rft.pub=Springer&rft.date=1987&rft.au=Jean-Claude+Martzloff&rft.au=S.S.+Wilson&rft.au=J.+Gernet&rft.au=J.+Dhombres&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Biggs-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Biggs_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Biggs_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggs1979" class="citation journal cs1">Biggs, N. L. (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2879%2990074-0">"The roots of combinatorics"</a>. <i>Historia Math</i>. <b>6</b> (2): 109–136. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2879%2990074-0">10.1016/0315-0860(79)90074-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Math.&rft.atitle=The+roots+of+combinatorics&rft.volume=6&rft.issue=2&rft.pages=109-136&rft.date=1979&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2879%2990074-0&rft.aulast=Biggs&rft.aufirst=N.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252879%252990074-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYadegari1980" class="citation journal cs1">Yadegari, Mohammad (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2880%2990004-X">"The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathematics"</a>. <i>Historia Mathematica</i>. <b>7</b> (4): 401–406. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2880%2990004-X">10.1016/0315-0860(80)90004-X</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+Binomial+Theorem%3A+A+Widespread+Concept+in+Medieval+Islamic+Mathematics&rft.volume=7&rft.issue=4&rft.pages=401-406&rft.date=1980&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2880%2990004-X&rft.aulast=Yadegari&rft.aufirst=Mohammad&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252880%252990004-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2015" class="citation journal cs1">Stillwell, John (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-2015-01491-6">"<i>Taming the unknown. A history of algebra ...</i> by Victor J. Katz and Karen Hunger Parshall"</a>. <i>Bulletin of the American Mathematical Society</i> (Book review). <b>52</b> (4): 725–731. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-2015-01491-6">10.1090/S0273-0979-2015-01491-6</a></span>. p. 727: <q>However, algebra advanced in other respects. Around 1000, al-Karaji stated the binomial theorem</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Taming+the+unknown.+A+history+of+algebra+...+by+Victor+J.+Katz+and+Karen+Hunger+Parshall&rft.volume=52&rft.issue=4&rft.pages=725-731&rft.date=2015&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-2015-01491-6&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0273-0979-2015-01491-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRashed1994" class="citation book cs1">Rashed, Roshdi (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vSkClSvU_9AC&pg=PA62"><i>The Development of Arabic Mathematics: Between Arithmetic and Algebra</i></a>. Kluwer. p. 63. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2565-6" title="Special:BookSources/0-7923-2565-6"><bdi>0-7923-2565-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Development+of+Arabic+Mathematics%3A+Between+Arithmetic+and+Algebra&rft.pages=63&rft.pub=Kluwer&rft.date=1994&rft.isbn=0-7923-2565-6&rft.aulast=Rashed&rft.aufirst=Roshdi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvSkClSvU_9AC%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Karaji-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Karaji_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Karaji_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. 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Springer. pp. 318–319. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387950006" title="Special:BookSources/9780387950006"><bdi>9780387950006</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Lie+Groups+to+Differential+Equations&rft.pages=318-319&rft.pub=Springer&rft.date=2000&rft.isbn=9780387950006&rft.aulast=Olver&rft.aufirst=Peter+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsI2bAxgLMXYC%26pg%3DPA318&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivey2019" class="citation book cs1">Spivey, Michael Z. (2019). <i>The Art of Proving Binomial Identities</i>. CRC Press. p. 71. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1351215800" title="Special:BookSources/978-1351215800"><bdi>978-1351215800</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Proving+Binomial+Identities&rft.pages=71&rft.pub=CRC+Press&rft.date=2019&rft.isbn=978-1351215800&rft.aulast=Spivey&rft.aufirst=Michael+Z.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoverThomas2001" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2001-01-01). <i>Data Compression</i>. John Wiley & Sons, Inc. p. 320. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F0471200611.ch5">10.1002/0471200611.ch5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471200611" title="Special:BookSources/9780471200611"><bdi>9780471200611</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+Compression&rft.pages=320&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2001-01-01&rft_id=info%3Adoi%2F10.1002%2F0471200611.ch5&rft.isbn=9780471200611&rft.aulast=Cover&rft.aufirst=Thomas+M.&rft.au=Thomas%2C+Joy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Artin, <i>Algebra</i>, 2nd edition, Pearson, 2018, equation (4.7.11).</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://arquivopessoa.net/textos/224">"Arquivo Pessoa: Obra Édita - O binómio de Newton é tão belo como a Vénus de Milo"</a>. arquivopessoa.net.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Arquivo+Pessoa%3A+Obra+%C3%89dita+-+O+bin%C3%B3mio+de+Newton+%C3%A9+t%C3%A3o+belo+como+a+V%C3%A9nus+de+Milo.&rft.pub=arquivopessoa.net&rft_id=http%3A%2F%2Farquivopessoa.net%2Ftextos%2F224&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBag1966" class="citation journal cs1">Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". <i>Indian J. History Sci</i>. <b>1</b> (1): 68–74.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+J.+History+Sci&rft.atitle=Binomial+theorem+in+ancient+India&rft.volume=1&rft.issue=1&rft.pages=68-74&rft.date=1966&rft.aulast=Bag&rft.aufirst=Amulya+Kumar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1">Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). "(5) Binomial Coefficients". <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/concretemathemat00grah_444"><i>Concrete Mathematics</i></a></span> (2nd ed.). Addison Wesley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/concretemathemat00grah_444/page/n165">153</a>–256. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-55802-9" title="Special:BookSources/978-0-201-55802-9"><bdi>978-0-201-55802-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/17649857">17649857</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%285%29+Binomial+Coefficients&rft.btitle=Concrete+Mathematics&rft.pages=153-256&rft.edition=2nd&rft.pub=Addison+Wesley&rft.date=1994&rft_id=info%3Aoclcnum%2F17649857&rft.isbn=978-0-201-55802-9&rft.aulast=Graham&rft.aufirst=Ronald&rft.au=Knuth%2C+Donald&rft.au=Patashnik%2C+Oren&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconcretemathemat00grah_444&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Binomial_theorem&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Combinatorics" class="extiw" title="wikibooks:Combinatorics">Combinatorics</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Combinatorics/Binomial_Theorem" class="extiw" title="wikibooks:Combinatorics/Binomial Theorem">The Binomial Theorem</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomentsev2001" class="citation cs2">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Newton_binomial">"Newton binomial"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Newton+binomial&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Solomentsev&rft.aufirst=E.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNewton_binomial&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+theorem" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/BinomialTheorem/">Binomial Theorem</a> by <a href="/wiki/Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a>, and <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/BinomialTheoremStepByStep/">"Binomial Theorem (Step-by-Step)"</a> by Bruce Colletti and Jeff Bryant, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li> <li><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/InductiveProofOfBinomialTheorem">inductive proof of binomial theorem</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control 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Binomial theorem - Wikipedia