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id="mw-panel-toc-list"> <li 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">Početak</div> </a> </li> <li id="toc-Definicija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicija</span> </div> </a> <ul id="toc-Definicija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primjeri" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primjeri"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Primjeri</span> </div> </a> <ul id="toc-Primjeri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Osnovna_svojstva" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Osnovna_svojstva"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Osnovna svojstva</span> </div> </a> <ul id="toc-Osnovna_svojstva-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Približno_računanje_faktorijela" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Približno_računanje_faktorijela"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Približno računanje faktorijela</span> </div> </a> <ul id="toc-Približno_računanje_faktorijela-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Viši_faktorijeli" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Viši_faktorijeli"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Viši faktorijeli</span> </div> </a> <ul id="toc-Viši_faktorijeli-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dvostruki_faktorijel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dvostruki_faktorijel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Dvostruki faktorijel</span> </div> </a> <ul id="toc-Dvostruki_faktorijel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primorijel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primorijel"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Primorijel</span> </div> </a> <ul id="toc-Primorijel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lijevi_faktorijel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lijevi_faktorijel"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Lijevi faktorijel</span> </div> </a> <ul id="toc-Lijevi_faktorijel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vidi_još" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vidi_još"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Vidi još</span> </div> </a> <ul id="toc-Vidi_još-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Faktorijel</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="Idi na druge jezične varijante članka. Dostupan je na 80 jezika" > <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-80" 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">80 jezika</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/Fakulteit_(wiskunde)" title="Fakulteit (wiskunde) – afrikaans" lang="af" hreflang="af" data-title="Fakulteit (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%8B%E1%8A%AD%E1%89%B6%E1%88%AA%E1%8B%AB%E1%88%8D" title="ፋክቶሪያል – amharski" lang="am" hreflang="am" data-title="ፋክቶሪያል" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%A7%D9%85%D9%84%D9%8A" title="عاملي – arapski" lang="ar" hreflang="ar" data-title="عاملي" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Factorial" title="Factorial – asturijski" lang="ast" hreflang="ast" data-title="Factorial" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Faktorial" title="Faktorial – azerbajdžanski" lang="az" hreflang="az" data-title="Faktorial" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал – baškirski" lang="ba" hreflang="ba" data-title="Факториал" data-language-autonym="Башҡортса" data-language-local-name="baškirski" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%B0%D1%80%D1%8B%D1%8F%D0%BB" title="Фактарыял – bjeloruski" lang="be" hreflang="be" data-title="Фактарыял" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B5%D0%BB" title="Факториел – bugarski" lang="bg" hreflang="bg" data-title="Факториел" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8C%E0%A6%A3%E0%A6%BF%E0%A6%95" title="গৌণিক – bangla" lang="bn" hreflang="bn" data-title="গৌণিক" data-language-autonym="বাংলা" data-language-local-name="bangla" 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/Faktorijel" title="Faktorijel – bosanski" lang="bs" hreflang="bs" data-title="Faktorijel" data-language-autonym="Bosanski" data-language-local-name="bosanski" 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/Factorial" title="Factorial – katalonski" lang="ca" hreflang="ca" data-title="Factorial" data-language-autonym="Català" data-language-local-name="katalonski" 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/%D9%81%D8%A7%DA%A9%D8%AA%DB%86%D8%B1%DB%8C%DB%8E%D9%84" title="فاکتۆریێل – soranski kurdski" lang="ckb" hreflang="ckb" data-title="فاکتۆریێل" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" 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/Faktori%C3%A1l" title="Faktoriál – češki" lang="cs" hreflang="cs" data-title="Faktoriál" data-language-autonym="Čeština" data-language-local-name="češki" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал – čuvaški" lang="cv" hreflang="cv" data-title="Факториал" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fakultet_(matematik)" title="Fakultet (matematik) – danski" lang="da" hreflang="da" data-title="Fakultet (matematik)" data-language-autonym="Dansk" data-language-local-name="danski" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik) – njemački" lang="de" hreflang="de" data-title="Fakultät (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="njemački" 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%A0%CE%B1%CF%81%CE%B1%CE%B3%CE%BF%CE%BD%CF%84%CE%B9%CE%BA%CF%8C" title="Παραγοντικό – grčki" lang="el" hreflang="el" data-title="Παραγοντικό" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="dobri članak"><a href="https://en.wikipedia.org/wiki/Factorial" title="Factorial – engleski" lang="en" hreflang="en" data-title="Factorial" data-language-autonym="English" data-language-local-name="engleski" 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/Faktorialo" title="Faktorialo – esperanto" lang="eo" hreflang="eo" data-title="Faktorialo" 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/Factorial" title="Factorial – španjolski" lang="es" hreflang="es" data-title="Factorial" data-language-autonym="Español" data-language-local-name="španjolski" 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/Faktoriaal" title="Faktoriaal – estonski" lang="et" hreflang="et" data-title="Faktoriaal" data-language-autonym="Eesti" data-language-local-name="estonski" 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/Faktorial" title="Faktorial – baskijski" lang="eu" hreflang="eu" data-title="Faktorial" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84" title="فاکتوریل – perzijski" lang="fa" hreflang="fa" data-title="فاکتوریل" data-language-autonym="فارسی" data-language-local-name="perzijski" 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/Kertoma" title="Kertoma – finski" lang="fi" hreflang="fi" data-title="Kertoma" data-language-autonym="Suomi" data-language-local-name="finski" 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/Factorielle" title="Factorielle – francuski" lang="fr" hreflang="fr" data-title="Factorielle" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Factorial" title="Factorial – galicijski" lang="gl" hreflang="gl" data-title="Factorial" data-language-autonym="Galego" data-language-local-name="galicijski" 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%A2%D7%A6%D7%A8%D7%AA_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="עצרת (מתמטיקה) – hebrejski" lang="he" hreflang="he" data-title="עצרת (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebrejski" 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%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="क्रमगुणित – hindski" lang="hi" hreflang="hi" data-title="क्रमगुणित" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Faktori%C3%A1lis" title="Faktoriális – mađarski" lang="hu" hreflang="hu" data-title="Faktoriális" data-language-autonym="Magyar" data-language-local-name="mađarski" 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%96%D5%A1%D5%AF%D5%BF%D5%B8%D6%80%D5%AB%D5%A1%D5%AC" title="Ֆակտորիալ – armenski" lang="hy" hreflang="hy" data-title="Ֆակտորիալ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Factorial" title="Factorial – interlingua" lang="ia" hreflang="ia" data-title="Factorial" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Faktorial" title="Faktorial – iban" lang="iba" hreflang="iba" data-title="Faktorial" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktorial" title="Faktorial – indonezijski" lang="id" hreflang="id" data-title="Faktorial" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Faktorialo" title="Faktorialo – ido" lang="io" hreflang="io" data-title="Faktorialo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/A%C3%B0feldi" title="Aðfeldi – islandski" lang="is" hreflang="is" data-title="Aðfeldi" data-language-autonym="Íslenska" data-language-local-name="islandski" 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/Fattoriale" title="Fattoriale – talijanski" lang="it" hreflang="it" data-title="Fattoriale" data-language-autonym="Italiano" data-language-local-name="talijanski" 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/%E9%9A%8E%E4%B9%97" title="階乗 – japanski" lang="ja" hreflang="ja" data-title="階乗" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A4%E1%83%90%E1%83%A5%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90%E1%83%9A%E1%83%98" title="მათემატიკური ფაქტორიალი – gruzijski" lang="ka" hreflang="ka" data-title="მათემატიკური ფაქტორიალი" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал – kazaški" lang="kk" hreflang="kk" data-title="Факториал" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B2%97%E0%B3%81%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಕ್ರಮಗುಣಿತ – karnatački" lang="kn" hreflang="kn" data-title="ಕ್ರಮಗುಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="karnatački" 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/%EA%B3%84%EC%8A%B9_(%EC%88%98%ED%95%99)" title="계승 (수학) – korejski" lang="ko" hreflang="ko" data-title="계승 (수학)" data-language-autonym="한국어" data-language-local-name="korejski" 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/Factorialis" title="Factorialis – latinski" lang="la" hreflang="la" data-title="Factorialis" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fatorial" title="Fatorial – Lombard" lang="lmo" hreflang="lmo" data-title="Fatorial" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Faktorialas" title="Faktorialas – litavski" lang="lt" hreflang="lt" data-title="Faktorialas" data-language-autonym="Lietuvių" data-language-local-name="litavski" 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/Faktori%C4%81ls" title="Faktoriāls – latvijski" lang="lv" hreflang="lv" data-title="Faktoriāls" data-language-autonym="Latviešu" data-language-local-name="latvijski" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B5%D0%BB" title="Факториел – makedonski" lang="mk" hreflang="mk" data-title="Факториел" data-language-autonym="Македонски" data-language-local-name="makedonski" 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%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%97%E0%B5%81%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ക്രമഗുണിതം – malajalamski" lang="ml" hreflang="ml" data-title="ക്രമഗുണിതം" data-language-autonym="മലയാളം" data-language-local-name="malajalamski" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="क्रमगुणित – marathski" lang="mr" hreflang="mr" data-title="क्रमगुणित" data-language-autonym="मराठी" data-language-local-name="marathski" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Faktorial" title="Faktorial – malajski" lang="ms" hreflang="ms" data-title="Faktorial" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" 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/Faculteit_(wiskunde)" title="Faculteit (wiskunde) – nizozemski" lang="nl" hreflang="nl" data-title="Faculteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemski" 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/Fakultet_i_matematikk" title="Fakultet i matematikk – norveški nynorsk" lang="nn" hreflang="nn" data-title="Fakultet i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški 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/Fakultet_(matematikk)" title="Fakultet (matematikk) – norveški bokmål" lang="nb" hreflang="nb" data-title="Fakultet (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Factoriala" title="Factoriala – okcitanski" lang="oc" hreflang="oc" data-title="Factoriala" data-language-autonym="Occitan" data-language-local-name="okcitanski" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%8D%E0%A8%B0%E0%A8%AE%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਕ੍ਰਮਗੁਣਿਤ – pandžapski" lang="pa" hreflang="pa" data-title="ਕ੍ਰਮਗੁਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžapski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Silnia" title="Silnia – poljski" lang="pl" hreflang="pl" data-title="Silnia" data-language-autonym="Polski" data-language-local-name="poljski" 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/Fatorial" title="Fatorial – Piedmontese" lang="pms" hreflang="pms" data-title="Fatorial" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fatorial" title="Fatorial – portugalski" lang="pt" hreflang="pt" data-title="Fatorial" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Factorial" title="Factorial – rumunjski" lang="ro" hreflang="ro" data-title="Factorial" data-language-autonym="Română" data-language-local-name="rumunjski" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал – ruski" lang="ru" hreflang="ru" data-title="Факториал" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Fatturiali" title="Fatturiali – sicilijski" lang="scn" hreflang="scn" data-title="Fatturiali" data-language-autonym="Sicilianu" data-language-local-name="sicilijski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Faktorijel" title="Faktorijel – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Faktorijel" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%8F%E0%B6%BB%E0%B7%9D%E0%B6%B4%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ක්‍රමාරෝපිතය – sinhaleški" lang="si" hreflang="si" data-title="ක්‍රමාරෝපිතය" data-language-autonym="සිංහල" data-language-local-name="sinhaleški" 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/Factorial" title="Factorial – Simple English" lang="en-simple" hreflang="en-simple" data-title="Factorial" 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/Faktori%C3%A1l" title="Faktoriál – slovački" lang="sk" hreflang="sk" data-title="Faktoriál" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fakulteta_(funkcija)" title="Fakulteta (funkcija) – slovenski" lang="sl" hreflang="sl" data-title="Fakulteta (funkcija)" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Faktoriali" title="Faktoriali – albanski" lang="sq" hreflang="sq" data-title="Faktoriali" data-language-autonym="Shqip" data-language-local-name="albanski" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%98%D0%B5%D0%BB" title="Факторијел – srpski" lang="sr" hreflang="sr" data-title="Факторијел" data-language-autonym="Српски / srpski" data-language-local-name="srpski" 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/Fakultet_(matematik)" title="Fakultet (matematik) – švedski" lang="sv" hreflang="sv" data-title="Fakultet (matematik)" data-language-autonym="Svenska" data-language-local-name="švedski" 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%A4%E0%AF%8A%E0%AE%9F%E0%AE%B0%E0%AF%8D_%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D" title="தொடர் பெருக்கம் – tamilski" lang="ta" hreflang="ta" data-title="தொடர் பெருக்கம்" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9F%E0%B8%81%E0%B8%97%E0%B8%AD%E0%B9%80%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%A5" title="แฟกทอเรียล – tajlandski" lang="th" hreflang="th" data-title="แฟกทอเรียล" data-language-autonym="ไทย" data-language-local-name="tajlandski" 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/Paktoryal" title="Paktoryal – tagalog" lang="tl" hreflang="tl" data-title="Paktoryal" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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/Fakt%C3%B6riyel" title="Faktöriyel – turski" lang="tr" hreflang="tr" data-title="Faktöriyel" data-language-autonym="Türkçe" data-language-local-name="turski" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D1%96%D0%B0%D0%BB" title="Факторіал – ukrajinski" lang="uk" hreflang="uk" data-title="Факторіал" data-language-autonym="Українська" data-language-local-name="ukrajinski" 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%B9%D8%A7%D9%85%D9%84%DB%8C%DB%81" title="عاملیہ – urdski" lang="ur" hreflang="ur" data-title="عاملیہ" data-language-autonym="اردو" data-language-local-name="urdski" 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/Faktorial" title="Faktorial – uzbečki" lang="uz" hreflang="uz" data-title="Faktorial" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" 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/Giai_th%E1%BB%ABa" title="Giai thừa – vijetnamski" lang="vi" hreflang="vi" data-title="Giai thừa" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" 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/%E9%98%B6%E4%B9%98" title="阶乘 – wu kineski" lang="wuu" hreflang="wuu" data-title="阶乘" data-language-autonym="吴语" data-language-local-name="wu kineski" 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/%E9%9A%8E%E4%B9%98" title="階乘 – kineski" lang="zh" hreflang="zh" data-title="階乘" data-language-autonym="中文" data-language-local-name="kineski" 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/%E9%9A%8E%E4%B9%98" title="階乘 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="階乘" 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/%E9%9A%8E%E4%B9%98" title="階乘 – kantonski" lang="yue" hreflang="yue" data-title="階乘" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q120976#sitelinks-wikipedia" title="Poveznice na druge 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sakrij</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Izvor: Wikipedija</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hr" dir="ltr"><table class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"> <caption>Prvih 15 faktorijela </caption> <tbody><tr> <th><i>n</i> </th> <th><i>n</i>! </th></tr> <tr> <td>0</td> <td>1 </td></tr> <tr> <td>1</td> <td>1 </td></tr> <tr> <td>2</td> <td>2 </td></tr> <tr> <td>3</td> <td>6 </td></tr> <tr> <td>4</td> <td>24 </td></tr> <tr> <td>5</td> <td>120 </td></tr> <tr> <td>6</td> <td>720 </td></tr> <tr> <td>7</td> <td>5 040 </td></tr> <tr> <td>8</td> <td>40 320 </td></tr> <tr> <td>9</td> <td>362 880 </td></tr> <tr> <td>10</td> <td>3 628 800 </td></tr> <tr> <td>11</td> <td>39 916 800 </td></tr> <tr> <td>12</td> <td>479 001 600 </td></tr> <tr> <td>13</td> <td>6 227 020 800 </td></tr> <tr> <td>14</td> <td>87 178 291 200 </td></tr> <tr> <td>15</td> <td>1 307 674 368 000 </td></tr></tbody></table> <p>U <a href="/wiki/Matematika" title="Matematika">matematici</a>, <b>faktorijel</b> ili <b>faktorijela</b> <a href="/wiki/Prirodni_broj" title="Prirodni broj">prirodnog broja</a> <i>n</i> je umnožak svih prirodnih brojeva manjih ili jednakih <i>n</i>. </p><p>Faktorijeli se koriste u <a href="/wiki/Kombinatorika" title="Kombinatorika">kombinatorici</a>, <a href="/wiki/Algebra" title="Algebra">algebri</a> te <a href="/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza">matematičkoj analizi</a>. </p><p>Notaciju <i>n</i>! uveo je Christian Kramp 1808. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicija">Definicija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=1" title="Uredi odlomak: Definicija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=1" title="Uredi kôd odjeljka Definicija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definicija faktorijela za prirodne brojeve: </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!=1\cdot 2\cdot \ldots \cdot n=\prod _{i=1}^{n}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <mi>n</mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=1\cdot 2\cdot \ldots \cdot n=\prod _{i=1}^{n}i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4959b8c821a118896939e1140490644f9269969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.878ex; height:6.843ex;" alt="{\displaystyle n!=1\cdot 2\cdot \ldots \cdot n=\prod _{i=1}^{n}i}"></span></dd></dl> <p>Dodatno je definirano: </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 0!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22956a0fa255c6c9562eab440f8c23c2954a6cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle 0!=1}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Primjeri">Primjeri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=2" title="Uredi odlomak: Primjeri" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=2" title="Uredi kôd odjeljka Primjeri"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd></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 1!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69681ee2b8227ac3a072941b474bd19afebc626f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle 1!=1}"></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 2!=1\cdot 2=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2!=1\cdot 2=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fddda54ee081cfcc68d39cf49c7584b5ac26ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.173ex; height:2.176ex;" alt="{\displaystyle 2!=1\cdot 2=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 5!=1\cdot 2\cdot 3\cdot 4\cdot 5=120}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <mo>=</mo> <mn>120</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5!=1\cdot 2\cdot 3\cdot 4\cdot 5=120}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a8407c750e7db04da88d96ba928413b94a812c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.022ex; height:2.176ex;" alt="{\displaystyle 5!=1\cdot 2\cdot 3\cdot 4\cdot 5=120}"></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 80!=1\cdot 2\cdot 3\cdot \ldots \cdot 79\cdot 80\approx 7.1569457\cdot 10^{118}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>80</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <mn>79</mn> <mo>⋅</mo> <mn>80</mn> <mo>≈</mo> <mn>7.1569457</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>118</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 80!=1\cdot 2\cdot 3\cdot \ldots \cdot 79\cdot 80\approx 7.1569457\cdot 10^{118}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc46f5d34fad4f025d942615c39b80e47e420466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:45.073ex; height:2.676ex;" alt="{\displaystyle 80!=1\cdot 2\cdot 3\cdot \ldots \cdot 79\cdot 80\approx 7.1569457\cdot 10^{118}}"></span></dd> <dd></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Osnovna_svojstva">Osnovna svojstva</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=3" title="Uredi odlomak: Osnovna svojstva" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=3" title="Uredi kôd odjeljka Osnovna svojstva"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Za faktorijele vrijedi rekurzivna relacija: </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!=n\cdot (n-1)!\qquad ,n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="2em" /> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n\cdot (n-1)!\qquad ,n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810649c4ea54ff89cc748df80a6bbbe62594facc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.402ex; height:2.843ex;" alt="{\displaystyle n!=n\cdot (n-1)!\qquad ,n\geq 1}"></span>.</dd></dl> <p><br /> </p><p>Faktorijeli brojeva većih od 1 su uvijek parni brojevi jer svaki broj pomnožen brojem 2, tj. bilo kojim parnim brojem, je paran broj. </p> <div class="mw-heading mw-heading2"><h2 id="Približno_računanje_faktorijela"><span id="Pribli.C5.BEno_ra.C4.8Dunanje_faktorijela"></span>Približno računanje faktorijela</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=4" title="Uredi odlomak: Približno računanje faktorijela" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=4" title="Uredi kôd odjeljka Približno računanje faktorijela"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Faktorijel relativno brzo raste što može predstavljati praktično ograničenje kod njegovog računanja. Na primjer, faktorijel broja 70 prelazi sto znamenki. Stoga je korisno za velike <i>n</i> koristiti takozvanu <i>Stirlingovu formulu</i> za približno izračunavanje faktorijela: </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!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977732acd9d1c6bbe89c54fc956f3c16d2feade9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.19ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"></span></dd> <dd></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Viši_faktorijeli"><span id="Vi.C5.A1i_faktorijeli"></span>Viši faktorijeli</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=5" title="Uredi odlomak: Viši faktorijeli" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=5" title="Uredi kôd odjeljka Viši faktorijeli"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Viši faktorijeli se označavaju ponavljanjem znaka '!', gdje broj ponavljanja označava redni broj višeg faktorijela. </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!...!=n\cdot (n-m)!...!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!...!=n\cdot (n-m)!...!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86fffb880502ed5d7e0dd8b25faae616b5df5bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.443ex; height:2.843ex;" alt="{\displaystyle n!...!=n\cdot (n-m)!...!}"></span> gdje je m redni broj višeg faktorijela.</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 z!...!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>!</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z!...!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30426560ca65de13829cf3fc2d15eebd580648b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.176ex;" alt="{\displaystyle z!...!=1}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z<=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo><=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z<=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9104306fd9d7af3927d4e337eb9ca949d07589ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.157ex; height:2.176ex;" alt="{\displaystyle z<=1}"></span></dd> <dd></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Dvostruki_faktorijel">Dvostruki faktorijel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=6" title="Uredi odlomak: Dvostruki faktorijel" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=6" title="Uredi kôd odjeljka Dvostruki faktorijel"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Izraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle n!!}"></span> označava <b>dvostruki faktorijel</b> i odnosi se na faktorijel <a href="/wiki/Paran_broj" class="mw-redirect" title="Paran broj">parnih</a> i <a href="/wiki/Neparan_broj" class="mw-redirect" title="Neparan broj">neparnih brojeva</a>. </p><p>Definiraju se na ovaj način: </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!!=\left\{{\begin{matrix}1,\qquad \quad \ &&{\mbox{za }}n=0{\mbox{ ili }}n=1;\\n(n-2)!!&&{\mbox{za }}n\geq 2.\qquad \qquad \end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mspace width="2em" /> <mspace width="1em" /> <mtext> </mtext> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>za </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> ili </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>za </mtext> </mstyle> </mrow> <mi>n</mi> <mo>≥</mo> <mn>2.</mn> <mspace width="2em" /> <mspace width="2em" /> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!!=\left\{{\begin{matrix}1,\qquad \quad \ &&{\mbox{za }}n=0{\mbox{ ili }}n=1;\\n(n-2)!!&&{\mbox{za }}n\geq 2.\qquad \qquad \end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/091d9071dcfe6fc8335efebdafe952fb7e130340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.19ex; height:6.176ex;" alt="{\displaystyle n!!=\left\{{\begin{matrix}1,\qquad \quad \ &&{\mbox{za }}n=0{\mbox{ ili }}n=1;\\n(n-2)!!&&{\mbox{za }}n\geq 2.\qquad \qquad \end{matrix}}\right.}"></span></dd></dl> <p>Primjerice: </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 6!!=2\cdot 4\cdot 6=48}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> <mo>=</mo> <mn>48</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6!!=2\cdot 4\cdot 6=48}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae258a265cd77909410656c906de64d0360844f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.824ex; height:2.176ex;" alt="{\displaystyle 6!!=2\cdot 4\cdot 6=48}"></span></li></ul> <p>ili pak </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 7!!=1\cdot 3\cdot 5\cdot 7=105}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>7</mn> <mo>=</mo> <mn>105</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7!!=1\cdot 3\cdot 5\cdot 7=105}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1891f196a76c451326115f65a4fe3ae0a6154546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.828ex; height:2.176ex;" alt="{\displaystyle 7!!=1\cdot 3\cdot 5\cdot 7=105}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Primorijel">Primorijel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=7" title="Uredi odlomak: Primorijel" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=7" title="Uredi kôd odjeljka Primorijel"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ako imamo prirodni broj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}"></span> tada je <span class="mwe-math-element"><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> <mi mathvariant="normal">#</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/167c3481839df4ace6689a25e170c5e0c0d5551e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle n\#}"></span> umnožak <a href="/wiki/Prosti_broj" title="Prosti broj">prostih brojeva</a> koji ne premašuju <span class="mwe-math-element"><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>. Izraz <span class="mwe-math-element"><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> <mi mathvariant="normal">#</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/167c3481839df4ace6689a25e170c5e0c0d5551e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle n\#}"></span> naziva se <i>n</i> <b>primorijela</b>. </p><p>Preciznije, </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\#={\begin{cases}1&{\text{ako}}\ {\text{je}}\ n=0,\ 1\\(n-1)\#\times n&{\text{ako}}\ {\text{je}}\ n\ {\text{prost}}\\(n-1)\#&{\text{ako}}\ n\ {\text{nije}}\ {\text{prost}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi mathvariant="normal">#</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ako</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>je</mtext> </mrow> <mtext> </mtext> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">#</mi> <mo>×</mo> <mi>n</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ako</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>je</mtext> </mrow> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>prost</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">#</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ako</mtext> </mrow> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>nije</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>prost</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\#={\begin{cases}1&{\text{ako}}\ {\text{je}}\ n=0,\ 1\\(n-1)\#\times n&{\text{ako}}\ {\text{je}}\ n\ {\text{prost}}\\(n-1)\#&{\text{ako}}\ n\ {\text{nije}}\ {\text{prost}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8a1116742557fd534af2a6fc7924ed9c68385d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:41.154ex; height:8.509ex;" alt="{\displaystyle n\#={\begin{cases}1&{\text{ako}}\ {\text{je}}\ n=0,\ 1\\(n-1)\#\times n&{\text{ako}}\ {\text{je}}\ n\ {\text{prost}}\\(n-1)\#&{\text{ako}}\ n\ {\text{nije}}\ {\text{prost}}.\end{cases}}}"></span></dd></dl> <p>Primijetimo da ćemo se, ako je <span class="mwe-math-element"><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> složen, pri računanju <i>spuštati</i> dok ne dođemo do prvog prostog broja. Zatim ćemo taj prosti broj zapisati u umnošku i opet se dalje spuštati do prvog manjeg prostog broja, pa zatim množimo onaj veći prosti broj s idućim manjim, i opet se spuštamo, itd. Time ćemo u konačnici zaista dobiti umnožak prostih brojeva manjih od <span class="mwe-math-element"><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>. </p><p>Evo kako bi tekao račun za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73ad051c5be72add8393030bdb4f6f3a0c42b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.509ex;" alt="{\displaystyle 9\#}"></span>. </p><p>Dakle, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9\#=8\#=7\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>8</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>7</mn> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\#=8\#=7\#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd239b78bdc425ed6c00fa19795010f7fd26d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.492ex; height:2.509ex;" alt="{\displaystyle 9\#=8\#=7\#}"></span> i sada računamo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7\cdot 6\#=7\cdot 5\#=7\cdot 5\cdot 4\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>⋅</mo> <mn>6</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>7</mn> <mo>⋅</mo> <mn>5</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>7</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>4</mn> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7\cdot 6\#=7\cdot 5\#=7\cdot 5\cdot 4\#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579b466fe48ef02eb62adeffb2dee152b78cb131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.858ex; height:2.509ex;" alt="{\displaystyle 7\cdot 6\#=7\cdot 5\#=7\cdot 5\cdot 4\#}"></span> i tako dalje sve do <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7\cdot 5\cdot 3\cdot 2\cdot 1\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7\cdot 5\cdot 3\cdot 2\cdot 1\#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c61925b0d2e12bf520744e83424d8a71ccb60d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.465ex; height:2.509ex;" alt="{\displaystyle 7\cdot 5\cdot 3\cdot 2\cdot 1\#}"></span>. Kako je po definiciji <span class="mwe-math-element"><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\#=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\#=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/698cbf1dc8206fce25f73e9cb7abf9a251decfc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle 1\#=1}"></span> slijedi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9\#=210}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>210</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\#=210}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c3a4e557c424939b8654436c6aa107b714d9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.684ex; height:2.509ex;" alt="{\displaystyle 9\#=210}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Lijevi_faktorijel">Lijevi faktorijel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=8" title="Uredi odlomak: Lijevi faktorijel" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=8" title="Uredi kôd odjeljka Lijevi faktorijel"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hrvatska" title="Hrvatska">Hrvatski</a> matematičar <a href="/wiki/Svetozar_Kurepa" title="Svetozar Kurepa">Svetozar Kurepa</a> (1907. – 1993.) objavio je 1971. godine definiciju <b>lijevog faktorijela</b>: <span class="mwe-math-element"><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!+1!+...+(n-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>!</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo>!</mo> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle !n=0!+1!+...+(n-1)!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0b7787f8f494181e0cfff0138dc9fcd4aca955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.171ex; height:2.843ex;" alt="{\displaystyle !n=0!+1!+...+(n-1)!}"></span> te postavio hipotezu, koja je i dalje otvorena, da je najveći zajednički djelitelj od <span class="mwe-math-element"><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"> <mo>!</mo> <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/efaf7d115d7a907ca111c252671fb63169b47a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle !n}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> za sve <span class="mwe-math-element"><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\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span> broj 2.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=9" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=9" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Kurepa/%7C">Duro Kurepa</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Vidi_još"><span id="Vidi_jo.C5.A1"></span>Vidi još</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorijel&veaction=edit&section=10" title="Uredi odlomak: Vidi još" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorijel&action=edit&section=10" title="Uredi kôd odjeljka Vidi još"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Binomni_koeficijent" title="Binomni koeficijent">Binomni koeficijent</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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