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data-event-name="pinnable-header.vector-toc.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">dölj</button> </div> <ul class="vector-toc-contents" 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">Inledning</div> </a> </li> <li id="toc-Tärningskast" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tärningskast"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Tärningskast</span> </div> </a> <ul id="toc-Tärningskast-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hattproblemet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hattproblemet"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Hattproblemet</span> </div> </a> <ul id="toc-Hattproblemet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generell_metod" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generell_metod"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generell metod</span> </div> </a> <ul id="toc-Generell_metod-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Delområden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Delområden"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Delområden</span> </div> </a> <button aria-controls="toc-Delområden-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Växla underavsnittet Delområden</span> </button> <ul id="toc-Delområden-sublist" class="vector-toc-list"> <li id="toc-Partitionsteori" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partitionsteori"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Partitionsteori</span> </div> </a> <ul id="toc-Partitionsteori-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Källor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Källor"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Källor</span> </div> </a> <ul id="toc-Källor-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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Kombinatorik</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="Gå till en artikel på ett annat språk. Tillgänglig på 79 språk" > <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-79" 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">79 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%8A%90_%E1%8C%A5%E1%88%9D%E1%88%A8%E1%89%B5" title="ሥነ ጥምረት – amhariska" lang="am" hreflang="am" data-title="ሥነ ጥምረት" data-language-autonym="አማርኛ" data-language-local-name="amhariska" 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%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%AA%D9%88%D8%A7%D9%81%D9%8A%D9%82%D9%8A" title="تحليل توافيقي – arabiska" lang="ar" hreflang="ar" data-title="تحليل توافيقي" data-language-autonym="العربية" data-language-local-name="arabiska" 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/Combinatoria" title="Combinatoria – asturiska" lang="ast" hreflang="ast" data-title="Combinatoria" data-language-autonym="Asturianu" data-language-local-name="asturiska" 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/Kombinatorika" title="Kombinatorika – azerbajdzjanska" lang="az" hreflang="az" data-title="Kombinatorika" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzjanska" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%81%E0%A6%9A%E0%A7%8D%E0%A6%9B-%E0%A6%AC%E0%A6%BF%E0%A6%A8%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="গুচ্ছ-বিন্যাসতত্ত্ব – bengali" lang="bn" hreflang="bn" data-title="গুচ্ছ-বিন্যাসতত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D1%8B%D0%BA%D0%B0" title="Камбінаторыка – belarusiska" lang="be" hreflang="be" data-title="Камбінаторыка" data-language-autonym="Беларуская" data-language-local-name="belarusiska" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D1%8B%D0%BA%D0%B0" title="Камбінаторыка – belarusiska (tarasjkevitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Камбінаторыка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusiska (tarasjkevitsa)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – bulgariska" lang="bg" hreflang="bg" data-title="Комбинаторика" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – bosniska" lang="bs" hreflang="bs" data-title="Kombinatorika" data-language-autonym="Bosanski" data-language-local-name="bosniska" 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/Combinat%C3%B2ria" title="Combinatòria – katalanska" lang="ca" hreflang="ca" data-title="Combinatòria" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – tjuvasjiska" lang="cv" hreflang="cv" data-title="Комбинаторика" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – tjeckiska" lang="cs" hreflang="cs" data-title="Kombinatorika" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfuniadeg" title="Cyfuniadeg – walesiska" lang="cy" hreflang="cy" data-title="Cyfuniadeg" data-language-autonym="Cymraeg" data-language-local-name="walesiska" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kombinatorik" title="Kombinatorik – danska" lang="da" hreflang="da" data-title="Kombinatorik" data-language-autonym="Dansk" data-language-local-name="danska" 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/Kombinatorik" title="Kombinatorik – tyska" lang="de" hreflang="de" data-title="Kombinatorik" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kombinatoorika" title="Kombinatoorika – estniska" lang="et" hreflang="et" data-title="Kombinatoorika" data-language-autonym="Eesti" data-language-local-name="estniska" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%B4%CF%85%CE%B1%CF%83%CF%84%CE%B9%CE%BA%CE%AE" title="Συνδυαστική – grekiska" lang="el" hreflang="el" data-title="Συνδυαστική" data-language-autonym="Ελληνικά" data-language-local-name="grekiska" 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/Combinatorics" title="Combinatorics – engelska" lang="en" hreflang="en" data-title="Combinatorics" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Combinatoria" title="Combinatoria – spanska" lang="es" hreflang="es" data-title="Combinatoria" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kombinatoriko" title="Kombinatoriko – esperanto" lang="eo" hreflang="eo" data-title="Kombinatoriko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Konbinatoria" title="Konbinatoria – baskiska" lang="eu" hreflang="eu" data-title="Konbinatoria" data-language-autonym="Euskara" data-language-local-name="baskiska" 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%AA%D8%B1%DA%A9%DB%8C%D8%A8%DB%8C%D8%A7%D8%AA" title="ترکیبیات – persiska" lang="fa" hreflang="fa" data-title="ترکیبیات" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Combinatoire" title="Combinatoire – franska" lang="fr" hreflang="fr" data-title="Combinatoire" data-language-autonym="Français" data-language-local-name="franska" 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/Combinatoria" title="Combinatoria – galiciska" lang="gl" hreflang="gl" data-title="Combinatoria" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" title="組合數學 – gan" lang="gan" hreflang="gan" data-title="組合數學" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A1%B0%ED%95%A9%EB%A1%A0" title="조합론 – koreanska" lang="ko" hreflang="ko" data-title="조합론" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%A2%D5%AB%D5%B6%D5%A1%D5%BF%D5%B8%D6%80%D5%AB%D5%AF%D5%A1" title="Կոմբինատորիկա – armeniska" lang="hy" hreflang="hy" data-title="Կոմբինատորիկա" data-language-autonym="Հայերեն" data-language-local-name="armeniska" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%9A%E0%A4%AF-%E0%A4%B8%E0%A4%82%E0%A4%9A%E0%A4%AF" title="क्रमचय-संचय – hindi" lang="hi" hreflang="hi" data-title="क्रमचय-संचय" 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/Kombinatorika" title="Kombinatorika – kroatiska" lang="hr" hreflang="hr" data-title="Kombinatorika" data-language-autonym="Hrvatski" data-language-local-name="kroatiska" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Kombinatoriko" title="Kombinatoriko – ido" lang="io" hreflang="io" data-title="Kombinatoriko" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – indonesiska" lang="id" hreflang="id" data-title="Kombinatorika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" 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/Talningarfr%C3%A6%C3%B0i" title="Talningarfræði – isländska" lang="is" hreflang="is" data-title="Talningarfræði" data-language-autonym="Íslenska" data-language-local-name="isländska" 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/Combinatoria" title="Combinatoria – italienska" lang="it" hreflang="it" data-title="Combinatoria" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%9E%D7%91%D7%99%D7%A0%D7%98%D7%95%D7%A8%D7%99%D7%A7%D7%94" title="קומבינטוריקה – hebreiska" lang="he" hreflang="he" data-title="קומבינטוריקה" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%91%E1%83%98%E1%83%9C%E1%83%90%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98%E1%83%99%E1%83%90" title="კომბინატორიკა – georgiska" lang="ka" hreflang="ka" data-title="კომბინატორიკა" data-language-autonym="ქართული" data-language-local-name="georgiska" 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%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – kazakiska" lang="kk" hreflang="kk" data-title="Комбинаторика" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Konbinatwar" title="Konbinatwar – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Konbinatwar" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – kirgiziska" lang="ky" hreflang="ky" data-title="Комбинаторика" data-language-autonym="Кыргызча" data-language-local-name="kirgiziska" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Calculus_coniunctionibus" title="Calculus coniunctionibus – latin" lang="la" hreflang="la" data-title="Calculus coniunctionibus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – lettiska" lang="lv" hreflang="lv" data-title="Kombinatorika" data-language-autonym="Latviešu" data-language-local-name="lettiska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – litauiska" lang="lt" hreflang="lt" data-title="Kombinatorika" data-language-autonym="Lietuvių" data-language-local-name="litauiska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Combinatoria" title="Combinatoria – lingua franca nova" lang="lfn" hreflang="lfn" data-title="Combinatoria" data-language-autonym="Lingua Franca Nova" data-language-local-name="lingua franca nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – ungerska" lang="hu" hreflang="hu" data-title="Kombinatorika" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – makedonska" lang="mk" hreflang="mk" data-title="Комбинаторика" data-language-autonym="Македонски" data-language-local-name="makedonska" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kombinatorik" title="Kombinatorik – malajiska" lang="ms" hreflang="ms" data-title="Kombinatorik" data-language-autonym="Bahasa Melayu" data-language-local-name="malajiska" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA" title="Комбинаторик – mongoliska" lang="mn" hreflang="mn" data-title="Комбинаторик" data-language-autonym="Монгол" data-language-local-name="mongoliska" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%94%E1%80%BA%E1%80%98%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%94%E1%80%90%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%9B%E1%80%85%E1%80%BA" title="ကွန်ဘိုင်နတိုးရစ် – burmesiska" lang="my" hreflang="my" data-title="ကွန်ဘိုင်နတိုးရစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmesiska" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Combinatoriek" title="Combinatoriek – nederländska" lang="nl" hreflang="nl" data-title="Combinatoriek" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%84%E5%90%88%E3%81%9B%E6%95%B0%E5%AD%A6" title="組合せ数学 – japanska" lang="ja" hreflang="ja" data-title="組合せ数学" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kombinatorikk" title="Kombinatorikk – norskt bokmål" lang="nb" hreflang="nb" data-title="Kombinatorikk" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kombinatorikk" title="Kombinatorikk – nynorska" lang="nn" hreflang="nn" data-title="Kombinatorikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorska" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – uzbekiska" lang="uz" hreflang="uz" data-title="Kombinatorika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekiska" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kambinatuorix" title="Kambinatuorix – jamaikansk engelsk kreol" lang="jam" hreflang="jam" data-title="Kambinatuorix" data-language-autonym="Patois" data-language-local-name="jamaikansk engelsk kreol" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kombinatoryka" title="Kombinatoryka – polska" lang="pl" hreflang="pl" data-title="Kombinatoryka" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Combinat%C3%B3ria" title="Combinatória – portugisiska" lang="pt" hreflang="pt" data-title="Combinatória" data-language-autonym="Português" data-language-local-name="portugisiska" 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/Combinatoric%C4%83" title="Combinatorică – rumänska" lang="ro" hreflang="ro" data-title="Combinatorică" data-language-autonym="Română" data-language-local-name="rumänska" 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%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – ryska" lang="ru" hreflang="ru" data-title="Комбинаторика" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – jakutiska" lang="sah" hreflang="sah" data-title="Комбинаторика" data-language-autonym="Саха тыла" data-language-local-name="jakutiska" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – albanska" lang="sq" hreflang="sq" data-title="Kombinatorika" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Combinatorics" title="Combinatorics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Combinatorics" 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/Kombinatorika" title="Kombinatorika – slovakiska" lang="sk" hreflang="sk" data-title="Kombinatorika" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" 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/Kombinatorika" title="Kombinatorika – slovenska" lang="sl" hreflang="sl" data-title="Kombinatorika" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Комбинаторна математика – serbiska" lang="sr" hreflang="sr" data-title="Комбинаторна математика" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – serbokroatiska" lang="sh" hreflang="sh" data-title="Kombinatorika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatiska" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kombinatoriikka" title="Kombinatoriikka – finska" lang="fi" hreflang="fi" data-title="Kombinatoriikka" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kombinatorika" title="Kombinatorika – tagalog" lang="tl" hreflang="tl" data-title="Kombinatorika" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="சேர்வியல் (கணிதம்) – tamil" lang="ta" hreflang="ta" data-title="சேர்வியல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбинаторика – tatariska" lang="tt" hreflang="tt" data-title="Комбинаторика" data-language-autonym="Татарча / tatarça" data-language-local-name="tatariska" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%88%E0%B8%B1%E0%B8%94" title="คณิตศาสตร์เชิงการจัด – thailändska" lang="th" hreflang="th" data-title="คณิตศาสตร์เชิงการจัด" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kombinatorik" title="Kombinatorik – turkiska" lang="tr" hreflang="tr" data-title="Kombinatorik" data-language-autonym="Türkçe" data-language-local-name="turkiska" 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%9A%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Комбінаторика – ukrainska" lang="uk" hreflang="uk" data-title="Комбінаторика" data-language-autonym="Українська" data-language-local-name="ukrainska" 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%AA%D8%A7%D9%84%DB%8C%D9%81%DB%8C%D8%A7%D8%AA" title="تالیفیات – urdu" lang="ur" hreflang="ur" data-title="تالیفیات" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_h%E1%BB%8Dc_t%E1%BB%95_h%E1%BB%A3p" title="Toán học tổ hợp – vietnamesiska" lang="vi" hreflang="vi" data-title="Toán học tổ hợp" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" 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/%E7%BB%84%E5%90%88%E6%95%B0%E5%AD%A6" title="组合数学 – wu" lang="wuu" hreflang="wuu" data-title="组合数学" 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-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%9E%D7%91%D7%99%D7%A0%D7%90%D7%98%D7%90%D7%A8%D7%99%D7%A7" title="קאמבינאטאריק – jiddisch" lang="yi" hreflang="yi" data-title="קאמבינאטאריק" data-language-autonym="ייִדיש" data-language-local-name="jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" title="組合數學 – kantonesiska" lang="yue" hreflang="yue" data-title="組合數學" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Kuomb%C4%97natuor%C4%97ka" title="Kuombėnatuorėka – samogitiska" lang="sgs" hreflang="sgs" data-title="Kuombėnatuorėka" data-language-autonym="Žemaitėška" data-language-local-name="samogitiska" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BB%84%E5%90%88%E6%95%B0%E5%AD%A6" title="组合数学 – kinesiska" lang="zh" hreflang="zh" data-title="组合数学" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Kombinatorik" title="Kombinatorik – ibanska" lang="iba" hreflang="iba" data-title="Kombinatorik" data-language-autonym="Jaku Iban" data-language-local-name="ibanska" class="interlanguage-link-target"><span>Jaku Iban</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/Q76592#sitelinks-wikipedia" title="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div 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class="noprint">Från Wikipedia</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="sv" dir="ltr"><p><b>Kombinatorik</b> är den gren av <a href="/wiki/Matematik" title="Matematik">matematiken</a> som studerar kombinationer, permutationer och uppräkningar av element i mängder och de relationer som karakteriserar dessas egenskaper. Metoderna för detta är grundläggande i den <a href="/wiki/Diskret_matematik" title="Diskret matematik">diskreta matematiken</a>. Föregångsmän inom området är bland andra <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>, <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, <a href="/wiki/Pierre_R%C3%A9mond_de_Montmort" title="Pierre Rémond de Montmort">Pierre Rémond de Montmort</a>, <a href="/wiki/James_Stirling_(matematiker)" title="James Stirling (matematiker)">James Stirling</a> och bröderna <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jacob</a> och <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a>. </p><p>Ett enkelt exempel på ett kombinatoriskt problem är frågan, om hur många olika ordningsföljder det finns av en 52-korts <a href="/wiki/Kortlek" title="Kortlek">kortlek</a>. Lösningsmetoden är känd sedan 2500 år tillbaka och antalet följder är 52! (utläses "<a href="/wiki/Fakultet_(matematik)" title="Fakultet (matematik)">fakulteten</a> av 52" eller "52 fakultet"), alltså 52·51·50· ··· ·3·2·1 vilket är ungefär lika med 8·10<sup>67</sup>, eller en åtta följd av 67 nollor. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tärningskast"><span id="T.C3.A4rningskast"></span>Tärningskast</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=1" title="Redigera avsnitt: Tärningskast" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=1" title="Redigera avsnitts källkod: Tärningskast"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kombinatoriken utvecklades jämsides med <a href="/wiki/Sannolikhetsteori" title="Sannolikhetsteori">sannolikhetsteorin</a> och kom att initieras hösten 1654 av <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a> i samband med att en vän till honom, chevalier <a href="/wiki/De_M%C3%A9r%C3%A9" class="mw-redirect" title="De Méré">de Méré</a>, ställde frågor rörande bland annat tärningsspel. Exempelvis kan man, vid kast med ett godtyckligt antal tärningar, med hjälp av Pascals <a href="/wiki/Binomialkoefficient" title="Binomialkoefficient">binomialkoefficienter</a> och <a href="/wiki/Stirlingtal" title="Stirlingtal">Stirlingtal</a>, skriva antalet möjliga utfall, som en strukturerad summa. Om antalet tärningar är fyra fås följande: </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 6^{4}=1\cdot {\tbinom {6}{1}}\cdot 1!+7\cdot {\tbinom {6}{2}}\cdot 2!+6\cdot {\tbinom {6}{3}}\cdot 3!+1\cdot {\tbinom {6}{4}}\cdot 4!=6+210+720+360=1296}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>!</mo> <mo>+</mo> <mn>7</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>2</mn> <mo>!</mo> <mo>+</mo> <mn>6</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>3</mn> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>4</mn> <mo>!</mo> <mo>=</mo> <mn>6</mn> <mo>+</mo> <mn>210</mn> <mo>+</mo> <mn>720</mn> <mo>+</mo> <mn>360</mn> <mo>=</mo> <mn>1296</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6^{4}=1\cdot {\tbinom {6}{1}}\cdot 1!+7\cdot {\tbinom {6}{2}}\cdot 2!+6\cdot {\tbinom {6}{3}}\cdot 3!+1\cdot {\tbinom {6}{4}}\cdot 4!=6+210+720+360=1296}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8146c6de16f7091a7dc3484fa8b007d3a5dbca37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:81.956ex; height:3.509ex;" alt="{\displaystyle 6^{4}=1\cdot {\tbinom {6}{1}}\cdot 1!+7\cdot {\tbinom {6}{2}}\cdot 2!+6\cdot {\tbinom {6}{3}}\cdot 3!+1\cdot {\tbinom {6}{4}}\cdot 4!=6+210+720+360=1296}"></span>, </p><p>där koefficienterna 1, 7, 6 och 1 är <a href="/wiki/Stirlingtal" title="Stirlingtal">Stirlingtal av andra slaget</a>. För händelsen, att alla tärningar visar lika, en <a href="/wiki/Ordnat_par" title="Ordnat par">kvadrupel</a>, är således antalet utfallsmöjligheter lika med 6, för en <a href="/wiki/Trippel" title="Trippel">trippel</a> eller <a href="/wiki/Ordnat_par" title="Ordnat par">två par</a> är antalet 210, för exakt ett par är antalet 720 och för att alla tärningar visar olika är antalet 360. </p> <div class="mw-heading mw-heading2"><h2 id="Hattproblemet">Hattproblemet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=2" title="Redigera avsnitt: Hattproblemet" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=2" title="Redigera avsnitts källkod: Hattproblemet"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ett klassiskt problem härstammar från den franske matematikern <a href="/wiki/De_Montmort-tal" title="De Montmort-tal">de Montmort</a>. Problemet, som har en mängd olika formuleringar, kan uttryckas så här: N herrar, som skall bevista en bankett lämnar sina hattar på en hylla och när de går hem väljer de, av någon anledning, en hatt helt slumpmässigt. Den fråga de Montmort ställde sig var: På hur många olika sätt kan hattarna väljas utan att någon får rätt hatt? Han brevväxlade, i frågan, med matematikern <a href="/wiki/Bernoulli" title="Bernoulli">Nicholas Bernoulli</a> under åren 1710-1712 och de lyckades finna en beräkningsmetod. I det speciella fall då N = 5, blir det sökta antalet: </p><p><a href="/wiki/De_Montmort-tal" title="De Montmort-tal">M(5)</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 =5!-{\tbinom {5}{1}}\cdot 4!+{\tbinom {5}{2}}\cdot 3!-{\tbinom {5}{3}}\cdot 2!+{\tbinom {5}{4}}\cdot 1!-{\tbinom {5}{5}}\cdot 0!=120-120+60-20+5-1=44.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>5</mn> <mo>!</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>4</mn> <mo>!</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>3</mn> <mo>!</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>2</mn> <mo>!</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>!</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>⋅</mo> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>120</mn> <mo>−</mo> <mn>120</mn> <mo>+</mo> <mn>60</mn> <mo>−</mo> <mn>20</mn> <mo>+</mo> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>44.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =5!-{\tbinom {5}{1}}\cdot 4!+{\tbinom {5}{2}}\cdot 3!-{\tbinom {5}{3}}\cdot 2!+{\tbinom {5}{4}}\cdot 1!-{\tbinom {5}{5}}\cdot 0!=120-120+60-20+5-1=44.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196ac9d1d6f1bf7eb5048d26bbd83e6a3d07a835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:87.986ex; height:3.509ex;" alt="{\displaystyle =5!-{\tbinom {5}{1}}\cdot 4!+{\tbinom {5}{2}}\cdot 3!-{\tbinom {5}{3}}\cdot 2!+{\tbinom {5}{4}}\cdot 1!-{\tbinom {5}{5}}\cdot 0!=120-120+60-20+5-1=44.}"></span> </p><p>Som synes är de två första termerna överflödiga, men tas med här av symmetriskäl. Talen har fått namn efter <a href="/wiki/De_Montmort-tal" title="De Montmort-tal">de Montmort</a> och för N = 1, 2, 3, 4, 5, 6.... är dessa M = 0, 1, 2, 9, 44, 265,.... Det visar sig att kvoten M/N! snabbt går mot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fa597122b7f406406cddde9f30d0dab2d33753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.416ex; height:2.676ex;" alt="{\displaystyle e^{-1}}"></span> då N växer. </p> <div class="mw-heading mw-heading2"><h2 id="Generell_metod">Generell metod</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=3" title="Redigera avsnitt: Generell metod" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=3" title="Redigera avsnitts källkod: Generell metod"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>För en systematisk behandling av kombinatoriska problem väljer man ofta en modell, där kulor under givna förutsättningar fördelas i lådor. Beroende på egenskaperna hos kulorna och lådorna, om de är särskiljbara eller inte särskiljbara, så varierar antalet <a href="/wiki/Tolvfaldiga_v%C3%A4gen" title="Tolvfaldiga vägen">fördelningsmöjligheter</a>. </p><p>I nedanstående tabell kallar vi antalet kulor för <i>n</i>, antalet lådor för <i>m</i> och <a href="/wiki/Stirlingtal" title="Stirlingtal">Stirlingtalet</a> av andra slaget för <i>S</i>(<i>n</i>, <i>m</i>). </p> <table class="wikitable"> <tbody><tr> <th>Kulorna är särskiljbara</th> <th>Lådorna är särskiljbara</th> <th>Lådor får vara tomma</th> <th>Antal möjligheter </th></tr> <tr> <td>Ja</td> <td>Ja</td> <td>Ja</td> <td><i>m</i><sup><i>n</i></sup> </td></tr> <tr> <td>Ja</td> <td>Ja</td> <td>Nej</td> <td><i>m</i>!·<i>S</i>(<i>n</i>, <i>m</i>) </td></tr> <tr> <td>Ja</td> <td>Nej</td> <td>Ja</td> <td><i>S</i>(<i>n</i>, 1) + <i>S</i>(<i>n</i>, 2) + ··· + <i>S</i>(<i>n</i>, <i>m</i>) </td></tr> <tr> <td>Ja</td> <td>Nej</td> <td>Nej</td> <td><i>S</i>(<i>n</i>, <i>m</i>) </td></tr> <tr> <td>Nej</td> <td>Ja</td> <td>Ja</td> <td><span class="mwe-math-element"><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+m-1 \choose n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+m-1 \choose n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422bb13883de3ce0c8495a503694e72040c5576f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.7ex; height:6.176ex;" alt="{\displaystyle n+m-1 \choose n}"></span> </td></tr> <tr> <td>Nej</td> <td>Ja</td> <td>Nej</td> <td><span class="mwe-math-element"><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 m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1 \choose m-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311019c0d5368f3aaa311600ef2c8929b27c4636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.464ex; height:6.176ex;" alt="{\displaystyle n-1 \choose m-1}"></span> </td></tr> </tbody></table> <p>Centrala begrepp inom kombinatoriken är <a href="/wiki/Kombination_(matematik)" title="Kombination (matematik)">kombination</a>, <a href="/wiki/Permutation" title="Permutation">permutation</a> och <a href="/wiki/Dirichlets_l%C3%A5dprincip" title="Dirichlets lådprincip">Dirichlets lådprincip</a>. </p><p>Kombinatorikens verktyg är av stor betydelse inom <a href="/wiki/Datavetenskap" title="Datavetenskap">datavetenskap</a>, <a href="/wiki/Sannolikhetsl%C3%A4ra" class="mw-redirect" title="Sannolikhetslära">sannolikhetslära</a> och <a href="/wiki/Statistik" title="Statistik">statistik</a>. Med kombinatorikens hjälp kan man, utifrån givna fakta, uttala sig med säkerhet. Till exempel: Det finns helt säkert minst tio svenskar som är lika långa på mikrometern när (se exempel i <a href="/wiki/Dirichlets_l%C3%A5dprincip" title="Dirichlets lådprincip">Dirichlets lådprincip</a>). För en systematisk och mer fullständig beskrivning av de grundläggande kombinatoriska metoderna, se artikeln, "<a href="/wiki/Tolvfaldiga_v%C3%A4gen" title="Tolvfaldiga vägen">Tolvfaldiga vägen</a>". </p> <div class="mw-heading mw-heading2"><h2 id="Delområden"><span id="Delomr.C3.A5den"></span>Delområden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=4" title="Redigera avsnitt: Delområden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=4" title="Redigera avsnitts källkod: Delområden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Partitionsteori">Partitionsteori</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=5" title="Redigera avsnitt: Partitionsteori" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=5" title="Redigera avsnitts källkod: Partitionsteori"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:Partition3D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Partition3D.svg/150px-Partition3D.svg.png" decoding="async" width="150" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Partition3D.svg/225px-Partition3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Partition3D.svg/300px-Partition3D.svg.png 2x" data-file-width="432" data-file-height="324" /></a><figcaption>En <a href="/w/index.php?title=Planpartition&action=edit&redlink=1" class="new" title="Planpartition [inte skriven än]">planpartition</a>.</figcaption></figure> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Heltalspartition" title="Heltalspartition">Heltalspartition</a></dd></dl></div> <p>Partitionsteori undersöker problem relaterade till heltalspartitioner, och är nära relaterat till <a href="/w/index.php?title=Q-serie&action=edit&redlink=1" class="new" title="Q-serie [inte skriven än]">q-serier</a>, <a href="/wiki/Speciell_funktion" title="Speciell funktion">speciella funktioner</a> och <a href="/wiki/Ortogonala_polynom" title="Ortogonala polynom">ortogonala polynom</a>. Ursprungligen var partitionsteori ett delområde av <a href="/wiki/Talteori" title="Talteori">talteori</a> och <a href="/wiki/Analys" title="Analys">analys</a>, men betraktas numera som en del av kombinatorik eller ett helt eget område. Den använder <a href="/w/index.php?title=Bijektivt_bevis&action=edit&redlink=1" class="new" title="Bijektivt bevis [inte skriven än]">bijektiva bevis</a> och flera verktyg från analys och <a href="/wiki/Analytisk_talteori" title="Analytisk talteori">analytisk talteori</a>, och har även samband med <a href="/wiki/Statistisk_mekanik" title="Statistisk mekanik">statistisk mekanik</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Källor"><span id="K.C3.A4llor"></span>Källor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kombinatorik&veaction=edit&section=6" title="Redigera avsnitt: Källor" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kombinatorik&action=edit&section=6" title="Redigera avsnitts källkod: Källor"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="tright portal" style="border:solid #aaa 1px; background:#f9f9f9; margin:0.5em 0 0.5em 0.5em; font-size:85%;"> <table style="background:transparent;" width="0"> <tbody><tr> <td><div style="position:relative; width:32px; height:28px; overflow:hidden"><div style="position:absolute; top:0px; left:0px; z-index:2;"> <span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/30px-Office-book.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/45px-Office-book.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/60px-Office-book.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></div> </td> <td>Den här artikeln ingår i boken: <br /><i><b><a href="/wiki/Wikipedia:B%C3%B6cker/Matematik" title="Wikipedia:Böcker/Matematik">Matematik</a></b></i>  </td></tr></tbody></table></div> <ul><li>Alan Tucker, Applied Combinatorics, John Wiley & Sons, New York, 1980.</li> <li>A. Hald, A History of Probability and Statistics and their Applications before 1750, Wiley, New York, 1990.</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; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Hämtad från ”<a dir="ltr" href="https://sv.wikipedia.org/w/index.php?title=Kombinatorik&oldid=54313485">https://sv.wikipedia.org/w/index.php?title=Kombinatorik&oldid=54313485</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorier" title="Wikipedia:Kategorier">Kategorier</a>: <ul><li><a href="/wiki/Kategori:Diskret_matematik" title="Kategori:Diskret matematik">Diskret matematik</a></li><li><a href="/wiki/Kategori:Kombinatorik" title="Kategori:Kombinatorik">Kombinatorik</a></li><li><a href="/wiki/Kategori:Sannolikhetsteori" title="Kategori:Sannolikhetsteori">Sannolikhetsteori</a></li><li><a href="/wiki/Kategori:Statistik" title="Kategori:Statistik">Statistik</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Dold kategori: <ul><li><a href="/wiki/Kategori:Artiklar_som_finns_i_bokformat" title="Kategori:Artiklar som finns i bokformat">Artiklar som finns i bokformat</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"> Sidan redigerades senast den 6 januari 2024 kl. 16.58.</li> <li id="footer-info-copyright">Wikipedias text är tillgänglig under licensen <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.sv">Creative Commons Erkännande-dela-lika 4.0 Unported</a>. 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