Binómio de Newton – Wikipédia, a enciclopédia livre

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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Notação_e_fórmula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notação_e_fórmula"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notação e fórmula</span> </div> </a> <ul id="toc-Notação_e_fórmula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_triângulo_de_Pascal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#O_triângulo_de_Pascal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>O triângulo de Pascal</span> </div> </a> <ul id="toc-O_triângulo_de_Pascal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Demonstração_do_teorema_do_Binômio_de_Newton" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Demonstração_do_teorema_do_Binômio_de_Newton"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Demonstração do teorema do Binômio de Newton</span> </div> </a> <ul id="toc-Demonstração_do_teorema_do_Binômio_de_Newton-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aplicações" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aplicações"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Aplicações</span> </div> </a> <ul id="toc-Aplicações-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <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="Alternar o índice" > <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">Alternar o índice</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">Binómio de Newton</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="Ir para um artigo noutra língua. Disponível em 70 línguas" > <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 línguas</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 — africanês" lang="af" hreflang="af" data-title="Binomiaalstelling" data-language-autonym="Afrikaans" data-language-local-name="africanês" 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 — árabe" lang="ar" hreflang="ar" data-title="مبرهنة ذات الحدين" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</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 — bielorrusso" lang="be" hreflang="be" data-title="Біном Ньютана" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" 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 — búlgaro" lang="bg" hreflang="bg" data-title="Нютонов бином" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</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-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 — bengalês" lang="bn" hreflang="bn" data-title="দ্বিপদী উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Binomna_teorema" title="Binomna teorema — bósnio" lang="bs" hreflang="bs" data-title="Binomna teorema" data-language-autonym="Bosanski" data-language-local-name="bósnio" 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 — catalão" lang="ca" hreflang="ca" data-title="Binomi de Newton" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%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 — curdo central" lang="ckb" hreflang="ckb" data-title="کراوەی دوو تێرمی" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Binomick%C3%A1_v%C4%9Bta" title="Binomická věta — checo" lang="cs" hreflang="cs" data-title="Binomická věta" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Binomischer_Lehrsatz" title="Binomischer Lehrsatz — alemão" lang="de" hreflang="de" data-title="Binomischer Lehrsatz" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CF%89%CE%BD%CF%85%CE%BC%CE%B9%CE%BA%CF%8C_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Διωνυμικό θεώρημα — grego" lang="el" hreflang="el" data-title="Διωνυμικό θεώρημα" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Binomial_theorem" title="Binomial theorem — inglês" lang="en" hreflang="en" data-title="Binomial theorem" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_del_binomio" title="Teorema del binomio — espanhol" lang="es" hreflang="es" data-title="Teorema del binomio" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Newtoni_binoomvalem" title="Newtoni binoomvalem — estónio" lang="et" hreflang="et" data-title="Newtoni binoomvalem" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Newtonen_binomio" title="Newtonen binomio — basco" lang="eu" hreflang="eu" data-title="Newtonen binomio" data-language-autonym="Euskara" data-language-local-name="basco" 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 — persa" lang="fa" hreflang="fa" data-title="بسط دوجمله‌ای" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Binomilause" title="Binomilause — finlandês" lang="fi" hreflang="fi" data-title="Binomilause" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Formule_du_bin%C3%B4me_de_Newton" title="Formule du binôme de Newton — francês" lang="fr" hreflang="fr" data-title="Formule du binôme de Newton" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Binomisk_Formeln" title="Binomisk Formeln — frísio setentrional" lang="frr" hreflang="frr" data-title="Binomisk Formeln" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</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 — galego" lang="gl" hreflang="gl" data-title="Teorema do binomio" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%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 — hebraico" lang="he" hreflang="he" data-title="הבינום של ניוטון" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AA%E0%A4%A6_%E0%A4%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 — croata" lang="hr" hreflang="hr" data-title="Binomni poučak" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Binomi%C3%A1lis_t%C3%A9tel" title="Binomiális tétel — húngaro" lang="hu" hreflang="hu" data-title="Binomiális tétel" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</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 — arménio" lang="hy" hreflang="hy" data-title="Նյուտոնի երկանդամ" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_binomial" title="Teorema binomial — indonésio" lang="id" hreflang="id" data-title="Teorema binomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" 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 — islandês" lang="is" hreflang="is" data-title="Tvíliðuregla" data-language-autonym="Íslenska" data-language-local-name="islandês" 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 — italiano" lang="it" hreflang="it" data-title="Teorema binomiale" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%AE%9A%E7%90%86" title="二項定理 — japonês" lang="ja" hreflang="ja" data-title="二項定理" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</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 — cazaque" lang="kk" hreflang="kk" data-title="Ньютон биномы" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</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-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 — canarim" lang="kn" hreflang="kn" data-title="ದ್ವಿಪದ ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarim" class="interlanguage-link-target"><span>ಕನ್ನಡ</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 — coreano" lang="ko" hreflang="ko" data-title="이항 정리" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theorema_binomiale" title="Theorema binomiale — latim" lang="la" hreflang="la" data-title="Theorema binomiale" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</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 — lituano" lang="lt" hreflang="lt" data-title="Binomo formulė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</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 — letão" lang="lv" hreflang="lv" data-title="Ņūtona binoms" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</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 — macedónio" lang="mk" hreflang="mk" data-title="Биномна теорема" data-language-autonym="Македонски" data-language-local-name="macedónio" 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 — malaiala" lang="ml" hreflang="ml" data-title="ദ്വിപദപ്രമേയം" data-language-autonym="മലയാളം" data-language-local-name="malaiala" 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 — malaio" lang="ms" hreflang="ms" data-title="Teorem binomial" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" 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 — neerlandês" lang="nl" hreflang="nl" data-title="Binomium van Newton" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Binomialformel" title="Binomialformel — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Binomialformel" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Binomialformel" title="Binomialformel — norueguês bokmål" lang="nb" hreflang="nb" data-title="Binomialformel" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dwumian_Newtona" title="Dwumian Newtona — polaco" lang="pl" hreflang="pl" data-title="Dwumian Newtona" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</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-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Binomul_lui_Newton" title="Binomul lui Newton — romeno" lang="ro" hreflang="ro" data-title="Binomul lui Newton" data-language-autonym="Română" data-language-local-name="romeno" 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 — russo" lang="ru" hreflang="ru" data-title="Бином Ньютона" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binomna_teorema" title="Binomna teorema — servo-croata" lang="sh" hreflang="sh" data-title="Binomna teorema" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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-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 — cingalês" lang="si" hreflang="si" data-title="ද්විපද ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="cingalês" 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 — eslovaco" lang="sk" hreflang="sk" data-title="Binomická veta" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</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 — albanês" lang="sq" hreflang="sq" data-title="Katrori i Binomit" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Биномна теорема — sérvio" lang="sr" hreflang="sr" data-title="Биномна теорема" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Binomialsatsen" title="Binomialsatsen — sueco" lang="sv" hreflang="sv" data-title="Binomialsatsen" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%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 — tâmil" lang="ta" hreflang="ta" data-title="ஈருறுப்புத் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</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 — tajique" lang="tg" hreflang="tg" data-title="Биноми Хайём-Нютон" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" class="interlanguage-link-target"><span>Тоҷикӣ</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 — tailandês" lang="th" hreflang="th" data-title="ทฤษฎีบททวินาม" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teoremang_binomial" title="Teoremang binomial — tagalo" lang="tl" hreflang="tl" data-title="Teoremang binomial" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</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 — turco" lang="tr" hreflang="tr" data-title="Binom açılımı" data-language-autonym="Türkçe" data-language-local-name="turco" 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 — ucraniano" lang="uk" hreflang="uk" data-title="Біном Ньютона" data-language-autonym="Українська" data-language-local-name="ucraniano" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Nyuton_binomi" title="Nyuton binomi — usbeque" lang="uz" hreflang="uz" data-title="Nyuton binomi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 — vietnamita" lang="vi" hreflang="vi" data-title="Định lý nhị thức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</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 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 — chinês" lang="zh" hreflang="zh" data-title="二项式定理" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</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-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 — cantonês" lang="yue" hreflang="yue" data-title="二項式定理" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div 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.tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Esta página <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">cita fontes</a>, mas que <b><a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">não cobrem</a> todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=Bin%C3%B3mio+de+Newton">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Bin%C3%B3mio+de+Newton&oq=Bin%C3%B3mio+de+Newton">notícias</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Bin%C3%B3mio+de+Newton">livros</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Bin%C3%B3mio+de+Newton">acadêmico</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Abril de 2012</span>)</i></small></div></td></tr></tbody></table> <p>Em <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a>, <span class="no-conversion"><span lang="pt-pt"><b>binómio de Newton</b></span> <sup>(<a href="/wiki/Portugu%C3%AAs_europeu" title="Português europeu">português europeu</a>)</sup> ou <span lang="pt-br"><b>binômio de Newton</b></span> <sup>(<a href="/wiki/Portugu%C3%AAs_brasileiro" title="Português brasileiro">português brasileiro</a>)</sup></span> permite escrever na forma <a href="/wiki/Base_can%C3%B4nica" title="Base canônica">canônica</a> o <a href="/wiki/Polin%C3%B3mio" class="mw-redirect" title="Polinómio">polinómio</a> correspondente à potência de um <a href="/wiki/Polin%C3%B3mio" class="mw-redirect" title="Polinómio">binómio</a>. O nome é dado em homenagem ao <a href="/wiki/F%C3%ADsico" title="Físico">físico</a> e <a href="/wiki/Matem%C3%A1tico" title="Matemático">matemático</a> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>. Entretanto, deve-se salientar que o Binômio de Newton não foi o objeto de estudos de <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>. Na verdade, o que <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> estudou foram regras que valem para <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+b)^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdfac89abdc81ad9084425d7401403b48d41e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:2.843ex;" alt="{\displaystyle (a+b)^{n}}" /></span> quando o expoente <b>n</b> é fracionário ou inteiro negativo, o que leva ao estudo de <a href="/wiki/S%C3%A9rie_(matem%C3%A1tica)" title="Série (matemática)">séries infinitas</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> </p><p>Casos particulares do Binômio de Newton são: </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 {\left(x+y\right)}^{1}=x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{1}=x+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74171ae31ed24afeed11695c92bd0ec46944b9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.613ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{1}=x+y}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left(x+y\right)}^{2}=x^{2}+2xy+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{2}=x^{2}+2xy+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b0ee74ae4144178925f9e84253616552b0ade5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.215ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{2}=x^{2}+2xy+y^{2}}" /></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notação_e_fórmula"><span id="Nota.C3.A7.C3.A3o_e_f.C3.B3rmula"></span>Notação e fórmula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&veaction=edit&section=1" title="Editar secção: Notação e fórmula" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&action=edit&section=1" title="Editar código-fonte da secção: Notação e fórmula"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O <a href="/wiki/Teorema" title="Teorema">teorema</a> do binômio de Newton se escreve como segue: </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 {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94ea38511415be07c2b1d123ece7781f541020f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.942ex; height:7.009ex;" alt="{\displaystyle {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}" /></span></dd></dl> <p>Os coeficientes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}" /></span> são chamados <a href="/wiki/Coeficientes_binomiais" class="mw-redirect" title="Coeficientes binomiais">coeficientes binomiais</a> e são definidos como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <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 {n \choose k}={\frac {n!}{k!(n-k)!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7e3b71232c18fc62a98583239c38a3cd5fb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.158ex; height:6.343ex;" alt="{\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}},}" /></span> onde <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> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> são inteiros, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/621f658bb51d7caac329d29e9bf435361813777f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.704ex; height:2.343ex;" alt="{\displaystyle k\leq n}" /></span> e <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\times 2\times \ldots x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mo>…</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x!=1\times 2\times \ldots x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5797bc3679be76616a3f3f9dab291b5773d52018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.521ex; height:2.176ex;" alt="{\displaystyle x!=1\times 2\times \ldots x}" /></span> é o <a href="/wiki/Fatorial" title="Fatorial">fatorial</a> de x.</dd></dl> <p>O coeficiente binomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}" /></span> corresponde, em <a href="/wiki/An%C3%A1lise_combinat%C3%B3ria" class="mw-redirect" title="Análise combinatória">análise combinatória</a>, ao número de combinações de <i>n</i> elementos agrupados <i>k</i> a <i>k</i>. </p> <div class="mw-heading mw-heading2"><h2 id="O_triângulo_de_Pascal"><span id="O_tri.C3.A2ngulo_de_Pascal"></span>O triângulo de Pascal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&veaction=edit&section=2" title="Editar secção: O triângulo de Pascal" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&action=edit&section=2" title="Editar código-fonte da secção: O triângulo de Pascal"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r69236695">.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}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span> Ver artigo principal: <a href="/wiki/Tri%C3%A2ngulo_de_Pascal" title="Triângulo de Pascal">Triângulo de Pascal</a></div> <p>Um <a href="/wiki/Algoritmo" title="Algoritmo">algoritmo</a> simples para calcular os coeficientes binomiais é o <a href="/wiki/Tri%C3%A2ngulo_de_Pascal" title="Triângulo de Pascal">triângulo de Pascal</a>. </p><p>O <b>triângulo de Pascal</b> é um <a href="/wiki/Tri%C3%A2ngulo" title="Triângulo">triângulo</a> numérico infinito formado por <a href="/wiki/Coeficiente_binomial" title="Coeficiente binomial">coeficientes binomiais</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 {\begin{matrix}{n \choose k}\end{matrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <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> </mtd> </mtr> </mtable> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{n \choose k}\end{matrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1f7b5fc6a01ab1d70c13e43c13e95218a7fbfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.817ex; margin-bottom: -0.187ex; width:4.514ex; height:3.176ex;" alt="{\displaystyle {\begin{matrix}{n \choose k}\end{matrix}},}" /></span> onde <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> representa o número da linha (posição vertical) e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> representa o número da coluna (posição horizontal). </p><p>A construção do triângulo faz-se de forma que cada elemento do triângulo de Pascal seja igual à soma dos elementos imediatamente acima e à direita com o elemento imediatamente acima e à esquerda. O elemento da primeira linha e primeira coluna é <b>1</b>. </p><p>O princípio do triângulo de Pascal é a <b><a href="/wiki/Rela%C3%A7%C3%A3o_de_Stifel" title="Relação de Stifel">relação de Stifel</a></b> também conhecida como <b>igualdade do triângulo de Pascal</b>: </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pascal_triangle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Pascal_triangle.png/500px-Pascal_triangle.png" decoding="async" width="440" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Pascal_triangle.png/960px-Pascal_triangle.png 1.5x" data-file-width="21750" data-file-height="1004" /></a><figcaption>O triângulo de Pascal.</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n-1 \choose k-1}+{n-1 \choose k}={n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n-1 \choose k-1}+{n-1 \choose k}={n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea31e451833c7d067a88050444072d374c455695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.392ex; height:6.176ex;" alt="{\displaystyle {n-1 \choose k-1}+{n-1 \choose k}={n \choose k}}" /></span></dd></dl> <p>Esta fórmula e o <a href="/wiki/Tri%C3%A2ngulo_de_Pascal" title="Triângulo de Pascal">triângulo de Pascal</a> são muitas vezes atribuídos a <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>, que os descreveu no <span style="white-space:nowrap;">século XVII</span>. Já eram, no entanto, conhecidos do matemático Chinês <a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a> no <span style="white-space:nowrap;">século XIII</span>. O matemático persa <a href="/wiki/Omar_Caiam" title="Omar Caiam">Omar Caiam</a>, pode ter sido o primeiro a descobrir. </p><p>Por exemplo, o desenvolvimento de diversos binômios através dessa técnica: </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 {\left(x+y\right)}^{2}=x^{2}y^{0}+2x^{1}y^{1}+x^{0}y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{2}=x^{2}y^{0}+2x^{1}y^{1}+x^{0}y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2faa0f93d765bc6613696d4a04f19147b46227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.927ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{2}=x^{2}y^{0}+2x^{1}y^{1}+x^{0}y^{2}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>3</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>1</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c92a7218c97e139893dd1b6d1ed46df176866c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.528ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left(x+y\right)}^{4}=x^{4}y^{0}+4x^{3}y^{1}+6x^{2}y^{2}+4x^{1}y^{3}+x^{0}y^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>4</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>1</mn> </mrow> </msup> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{4}=x^{4}y^{0}+4x^{3}y^{1}+6x^{2}y^{2}+4x^{1}y^{3}+x^{0}y^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d25cfb2563b35b45513529f444fbe0729985076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.777ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{4}=x^{4}y^{0}+4x^{3}y^{1}+6x^{2}y^{2}+4x^{1}y^{3}+x^{0}y^{4}.}" /></span><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Triangulo_de_Pascal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Triangulo_de_Pascal.svg/220px-Triangulo_de_Pascal.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Triangulo_de_Pascal.svg/330px-Triangulo_de_Pascal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Triangulo_de_Pascal.svg/440px-Triangulo_de_Pascal.svg.png 2x" data-file-width="2731" data-file-height="1972" /></a><figcaption>O triângulo de Pascal</figcaption></figure><br /></dd></dl> <p>Para resolvermos binômios do tipo (x+y)<sup>n</sup> é possível utilizar o triângulo de pascal, onde <i>n</i> é a linha reapresentada no triângulo (na imagem indo de 0 à 14). Para iniciar o processo utilizamos o primeiro (x) termo da esquerda para a direita: </p><p>(x+y)<sup>n</sup>= __x<sup>n</sup>___+__x<sup>(n-1)</sup>__x<sup>(n-2)</sup>+ ...+__x<sup>(n-n)</sup>__ </p><p>Agora seguindo o mesmo procedimento para o segundo termo (y), porém da direita para a esquerda: </p><p>(x+y)<sup>n</sup>=__x<sup>n</sup> y<sup>(n-n)</sup>+__x<sup>(n-1)</sup> y<sup>1</sup>+__x<sup>(n-2)</sup> y<sup>2</sup>+ ...+__x<sup>(n-n)</sup> y<sup>n</sup>. </p><p>Para sabermos os coeficientes deste binômio basta olhar, no triângulo de Pascal, a n-ésima linha e colocá-los na ordem em que se encontra. </p><p>Para isso, segue o seguinte exemplo: </p><p><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 {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>3</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>1</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c92a7218c97e139893dd1b6d1ed46df176866c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.528ex; height:3.343ex;" alt="{\displaystyle {\left(x+y\right)}^{3}=x^{3}y^{0}+3x^{2}y^{1}+3x^{1}y^{2}+x^{0}y^{3}}" /></span> </p><p>Podemos ver que os coeficientes correspondem aos da linha 3 do triângulo de Pascal. </p><p>Neste exemplo podemos verificar que os coeficientes são, consecutivamente, os valores da linha 3 do triângulo de Pascal. </p><p>Sendo assim teríamos para cada linha do triângulo de Pascal um binômio<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup>: </p> <table class="wikitable"> <tbody><tr> <td><b>n</b> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>(x+y)<sup>n</sup> </td></tr> <tr> <td><b>0</b> </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>1 </td></tr> <tr> <td><b>1</b> </td> <td>1 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>x+y </td></tr> <tr> <td><b>2</b> </td> <td>1 </td> <td>2 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>x<sup>2</sup>+2xy+y<sup>2</sup> </td></tr> <tr> <td><b>3</b> </td> <td>1 </td> <td>3 </td> <td>3 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td>x<sup>3</sup>+3x<sup>2</sup>y+3xy<sup>2</sup>+y<sup>3</sup> </td></tr> <tr> <td><b>4</b> </td> <td>1 </td> <td>4 </td> <td>6 </td> <td>4 </td> <td>1 </td> <td> </td> <td> </td> <td>x<sup>4</sup>+4x<sup>3</sup>y+6x<sup>2</sup>y<sup>2</sup>+4xy<sup>3</sup>+y<sup>4</sup> </td></tr> <tr> <td><b>5</b> </td> <td>1 </td> <td>5 </td> <td>10 </td> <td>10 </td> <td>5 </td> <td>1 </td> <td> </td> <td>x<sup>5</sup>+5x<sup>4</sup>y+10x<sup>3</sup>y<sup>2</sup>+10x<sup>2</sup>y<sup>3</sup>+5xy<sup>4</sup>+y<sup>5</sup> </td></tr> <tr> <td><b>6</b> </td> <td>1 </td> <td>6 </td> <td>15 </td> <td>20 </td> <td>15 </td> <td>6 </td> <td>1 </td> <td>x<sup>6</sup>+6x<sup>5</sup>y+15x<sup>4</sup>y<sup>2</sup>+20x<sup>3</sup>y<sup>3</sup>+15x<sup>2</sup>y<sup>4</sup>+6xy<sup>5</sup>+y<sup>6</sup> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Demonstração_do_teorema_do_Binômio_de_Newton"><span id="Demonstra.C3.A7.C3.A3o_do_teorema_do_Bin.C3.B4mio_de_Newton"></span>Demonstração do teorema do Binômio de Newton</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&veaction=edit&section=3" title="Editar secção: Demonstração do teorema do Binômio de Newton" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&action=edit&section=3" title="Editar código-fonte da secção: Demonstração do teorema do Binômio de Newton"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Tri%C3%A1ngulo_de_Pascal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Tri%C3%A1ngulo_de_Pascal.png/220px-Tri%C3%A1ngulo_de_Pascal.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/24/Tri%C3%A1ngulo_de_Pascal.png 1.5x" data-file-width="300" data-file-height="195" /></a><figcaption>⠀⠀⠀⠀⠀⠀</figcaption></figure> <p>Antes de começar, vale lembrar que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n-1}a_{k}=\sum _{k=1}^{n}a_{k-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n-1}a_{k}=\sum _{k=1}^{n}a_{k-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c97d6d624c81ffb9ed1c0fe7afe9d40598b2b4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.32ex; height:7.509ex;" alt="{\displaystyle \sum _{k=0}^{n-1}a_{k}=\sum _{k=1}^{n}a_{k-1}}" /></span> (1)</dd></dl> <p>Sejam <i>x</i>, <i>y</i> elementos de um <a href="/wiki/Anel_comutativo" title="Anel comutativo">anel comutativo</a>( <i>xy=yx</i>) e <i>n</i> um <a href="/wiki/Inteiro" class="mw-redirect" title="Inteiro">inteiro</a> não-negativo. </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 (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-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> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c80c2502413b225d48e79a475156a5f7b677e21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.942ex; height:7.009ex;" alt="{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}" /></span></dd></dl> <p>Demonstraremos por <a href="/wiki/Indu%C3%A7%C3%A3o_matem%C3%A1tica" title="Indução matemática">indução matemática</a>. </p> <dl><dd><dl><dd><i>Base</i>:</dd></dl></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0~,\qquad (x+y)^{0}=1={0 \choose 0}x^{0}y^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> <mspace width="2em"></mspace> <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> <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"> <mn>0</mn> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0~,\qquad (x+y)^{0}=1={0 \choose 0}x^{0}y^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e80f5faaaa9453734f72f8272099a63e98dff07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.646ex; height:6.176ex;" alt="{\displaystyle n=0~,\qquad (x+y)^{0}=1={0 \choose 0}x^{0}y^{0}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1~,\qquad (x+y)^{1}=x+y={1 \choose 0}x^{1}y^{0}+{1 \choose 1}x^{0}y^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> <mspace width="2em"></mspace> <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> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</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"> <mn>1</mn> <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>1</mn> </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"> <mn>1</mn> <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>0</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1~,\qquad (x+y)^{1}=x+y={1 \choose 0}x^{1}y^{0}+{1 \choose 1}x^{0}y^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c0a8007fbbfafa7de65c22e1f14177a8fa804b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.832ex; height:6.176ex;" alt="{\displaystyle n=1~,\qquad (x+y)^{1}=x+y={1 \choose 0}x^{1}y^{0}+{1 \choose 1}x^{0}y^{1}}" /></span></dd></dl> <dl><dd><dl><dd><i>Recorrência</i>:</dd></dl></dd></dl> <p>Seja <i>n</i> um inteiro maior ou igual a 1, mostraremos que a relação para <i>n</i> implica a relação para <i>n+1</i>: </p><p>Da hipótese de indução: </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 (x+y)^{n+1}=(x+y)\cdot \sum _{k=0}^{n}{n \choose k}x^{n-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> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=(x+y)\cdot \sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e77f28ba569cf0efce946c937c34ac2f09e8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.503ex; height:7.009ex;" alt="{\displaystyle (x+y)^{n+1}=(x+y)\cdot \sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k},}" /></span></dd></dl> <p>Por <a href="/wiki/Distributividade" title="Distributividade">distributividade</a> de produto sob a soma: </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 (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=0}^{n-1}{n \choose k}x^{n-k}y^{k}+y^{n+1}}"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <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> <mi>y</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> <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"> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=0}^{n-1}{n \choose k}x^{n-k}y^{k}+y^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ebd76f54d5e1043cd0114d2fc16dab276aea17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.026ex; height:7.509ex;" alt="{\displaystyle (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=0}^{n-1}{n \choose k}x^{n-k}y^{k}+y^{n+1}}" /></span></dd></dl> <p>Que pode ser reescrito usando (1): </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 (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=1}^{n}{n \choose k-1}x^{n-k+1}y^{k-1}+y^{n+1}}"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <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> <mi>y</mi> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=1}^{n}{n \choose k-1}x^{n-k+1}y^{k-1}+y^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b0f16c9ff4e427bdb7ce4df8547ec81e281b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.046ex; height:6.843ex;" alt="{\displaystyle (x+y)^{n+1}=x^{n+1}+x\cdot \sum _{k=1}^{n}{n \choose k}x^{n-k}y^{k}+y\cdot \sum _{k=1}^{n}{n \choose k-1}x^{n-k+1}y^{k-1}+y^{n+1}}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}\left\lbrack {{n} \choose {k}}+{{n} \choose {k-1}}\right\rbrack x^{n-k+1}y^{k}+y^{n+1}}"> <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> <msup> <mi>x</mi> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> <mo>]</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}\left\lbrack {{n} \choose {k}}+{{n} \choose {k-1}}\right\rbrack x^{n-k+1}y^{k}+y^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0235576afc11415220a148b0753baa1e5d7cf801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.27ex; height:6.843ex;" alt="{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}\left\lbrack {{n} \choose {k}}+{{n} \choose {k-1}}\right\rbrack x^{n-k+1}y^{k}+y^{n+1}}" /></span></dd></dl> <p>Usando a formula do <a href="/wiki/Tri%C3%A2ngulo_de_Pascal" title="Triângulo de Pascal">triângulo de Pascal</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}{{n+1} \choose k}~x^{n-k+1}y^{k}+y^{n+1}}"> <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> <msup> <mi>x</mi> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <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> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}{{n+1} \choose k}~x^{n-k+1}y^{k}+y^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad61d5f32f25c5746686e4e115c6d764587c69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.536ex; height:6.843ex;" alt="{\displaystyle (x+y)^{n+1}=x^{n+1}+\sum _{k=1}^{n}{{n+1} \choose k}~x^{n-k+1}y^{k}+y^{n+1}}" /></span></dd></dl> <p>Reagrupando o <a href="/wiki/Somat%C3%B3rio" title="Somatório">somatório</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{{n+1} \choose k}~x^{n-k+1}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 class="MJX-TeXAtom-ORD"> <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> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{{n+1} \choose k}~x^{n-k+1}y^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533bb9ae3f06b1dc9f8de896b2e65558d8c2e969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.727ex; height:7.509ex;" alt="{\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{{n+1} \choose k}~x^{n-k+1}y^{k}}" /></span></dd></dl> <p>E segue o resultado. </p> <div class="mw-heading mw-heading2"><h2 id="Aplicações"><span id="Aplica.C3.A7.C3.B5es"></span>Aplicações</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&veaction=edit&section=4" title="Editar secção: Aplicações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&action=edit&section=4" title="Editar código-fonte da secção: Aplicações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O binómio de Newton pode ser usado para derivar diversas expressões matemáticas, através da escolha adequada de <i>x</i> e <i>y</i>. Por exemplo: </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 0=(1-1)^{n}=\sum _{k=0}^{n}{n \choose k}(-1)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=(1-1)^{n}=\sum _{k=0}^{n}{n \choose k}(-1)^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc568ecd0f39425ade2171dc39e4613c65f8712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.979ex; height:7.009ex;" alt="{\displaystyle 0=(1-1)^{n}=\sum _{k=0}^{n}{n \choose k}(-1)^{k}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}=(1+1)^{n}=\sum _{k=0}^{n}{n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}=(1+1)^{n}=\sum _{k=0}^{n}{n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/800785e461dfa7dd4150c01899850cc65f9a54a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.329ex; height:7.009ex;" alt="{\displaystyle 2^{n}=(1+1)^{n}=\sum _{k=0}^{n}{n \choose k}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2n}=(1+1)^{2n}=\sum _{k=0}^{2n}{2n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <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"> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2n}=(1+1)^{2n}=\sum _{k=0}^{2n}{2n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/681fcd0911bc2779da90b59092421bd2392e4136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.135ex; height:7.509ex;" alt="{\displaystyle 2^{2n}=(1+1)^{2n}=\sum _{k=0}^{2n}{2n \choose k}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=[x+(1-x)]^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}(1-x)^{n-k}=\sum _{k=0}^{n}B_{k}^{n}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <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> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=[x+(1-x)]^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}(1-x)^{n-k}=\sum _{k=0}^{n}B_{k}^{n}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c996d7f14948f3df485a19e250f0d1322eb7903c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.264ex; height:7.009ex;" alt="{\displaystyle 1=[x+(1-x)]^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}(1-x)^{n-k}=\sum _{k=0}^{n}B_{k}^{n}(x),}" /></span> onde <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_{k}^{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{k}^{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df78becc07e9ab0bcff05aa63bacb54970008239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.121ex; height:3.009ex;" alt="{\displaystyle B_{k}^{n}(x)}" /></span> são os <a href="/wiki/Polin%C3%B3mios_de_Bernstein" title="Polinómios de Bernstein">polinómios de Bernstein</a>.</li></ul> <dl><dd>Recomendado:<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 {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94ea38511415be07c2b1d123ece7781f541020f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.942ex; height:7.009ex;" alt="{\displaystyle {\left(x+y\right)}^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&veaction=edit&section=5" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bin%C3%B3mio_de_Newton&action=edit&section=5" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tri%C3%A2ngulo_de_Pascal" title="Triângulo de Pascal">Triângulo de Pascal</a></li> <li><a href="/wiki/Combina%C3%A7%C3%A3o_(matem%C3%A1tica)" class="mw-redirect" title="Combinação (matemática)">Combinação</a></li> <li><a href="/wiki/Polin%C3%B4mio" class="mw-redirect" title="Polinômio">Polinômios</a></li></ul> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">GARBI, Gilberto G. O Romance das Equações Algébricas. Editora Livraria da Física. São Paulo, 2007. <a href="/wiki/Especial:Fontes_de_livros/8588325764" class="internal mw-magiclink-isbn">ISBN 85-88325-76-4</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-pascal-2009-1.pdf">«Pascal's triangle and the binomial theorem»</a> <span style="font-size:85%;">(PDF)</span>. <i>www.mathcentre.ac.uk</i><span class="reference-accessdate">. Consultado em 5 de dezembro de 2018</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ABin%C3%B3mio+de+Newton&rft.atitle=Pascal%99s+triangle+and+the+binomial+theorem&rft.genre=unknown&rft.jtitle=www.mathcentre.ac.uk&rft_id=http%3A%2F%2Fwww.mathcentre.ac.uk%2Fresources%2Fuploaded%2Fmc-ty-pascal-2009-1.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Binómio_de_Newton&oldid=69039617">https://pt.wikipedia.org/w/index.php?title=Binómio_de_Newton&oldid=69039617</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:Combinat%C3%B3ria" title="Categoria:Combinatória">Combinatória</a></li><li><a href="/wiki/Categoria:Isaac_Newton" title="Categoria:Isaac Newton">Isaac Newton</a></li><li><a href="/wiki/Categoria:Teoremas_em_%C3%A1lgebra" title="Categoria:Teoremas em álgebra">Teoremas em álgebra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!P%C3%A1ginas_que_usam_hiperliga%C3%A7%C3%B5es_m%C3%A1gicas_ISBN" title="Categoria:!Páginas que usam hiperligações mágicas ISBN">!Páginas que usam hiperligações mágicas ISBN</a></li><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_desde_abril_de_2012" title="Categoria:!Artigos que carecem de notas de rodapé desde abril de 2012">!Artigos que carecem de notas de rodapé desde abril de 2012</a></li><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_sem_indica%C3%A7%C3%A3o_de_tema" title="Categoria:!Artigos que carecem de notas de rodapé sem indicação de tema">!Artigos que carecem de notas de rodapé sem indicação de tema</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 23h43min de 21 de novembro de 2024.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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Binómio de Newton – Wikipédia, a enciclopédia livre