CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="mr" dir="ltr"> <head> <meta charset="UTF-8"> <title>क्रमगुणित - विकिपीडिया</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )mrwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["0\t1\t2\t3\t4\t5\t6\t7\t8\t9", "०\t१\t२\t३\t४\t५\t६\t७\t८\t९"],"wgDefaultDateFormat":"dmy","wgMonthNames":["","जानेवारी","फेब्रुवारी","मार्च","एप्रिल","मे","जून","जुलै","ऑगस्ट","सप्टेंबर","ऑक्टोबर","नोव्हेंबर","डिसेंबर"],"wgRequestId":"723dd7ba-5123-4278-9bf7-9a3c2fdd5100","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"क्रमगुणित","wgTitle":"क्रमगुणित","wgCurRevisionId":2176557,"wgRevisionId":2176557,"wgArticleId":295936,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Pages with non-numeric formatnum arguments","गणित","अंकगणित"],"wgPageViewLanguage":"mr","wgPageContentLanguage":"mr","wgPageContentModel":"wikitext","wgRelevantPageName":"क्रमगुणित", "wgRelevantArticleId":295936,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"mr","pageLanguageDir":"ltr","pageVariantFallbacks":"mr"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":9000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q120976","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics": true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.page.media","site","mediawiki.page.ready","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.refToolbar","ext.gadget.ProtectionIndicators","ext.gadget.AdvancedSiteNotices","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups", "ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.shortUrl","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=mr&modules=ext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=mr&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=mr&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="क्रमगुणित - विकिपीडिया"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//mr.m.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"> <link rel="alternate" type="application/x-wiki" title="लिहा" href="/w/index.php?title=%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&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="विकिपीडिया (mr)"> <link rel="EditURI" type="application/rsd+xml" href="//mr.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" 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"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.mr"> <link rel="alternate" type="application/atom+xml" title="विकिपीडिया अ‍ॅटम रसद (Atom Feed)" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%85%E0%A4%B2%E0%A5%80%E0%A4%95%E0%A4%A1%E0%A5%80%E0%A4%B2_%E0%A4%AC%E0%A4%A6%E0%A4%B2&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-क्रमगुणित rootpage-क्रमगुणित skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> सुचालन </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%E0%A4%AE%E0%A5%81%E0%A4%96%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A0" title="मुखपृष्ठाला भेट द्या [z]" accesskey="z"><span>मुखपृष्ठ</span></a></li><li id="n-धूळपाटी" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%A7%E0%A5%82%E0%A4%B3%E0%A4%AA%E0%A4%BE%E0%A4%9F%E0%A5%80/%E0%A4%95%E0%A5%87%E0%A4%B5%E0%A4%B3_%E0%A4%AE%E0%A4%B0%E0%A4%BE%E0%A4%A0%E0%A5%80"><span>धूळपाटी</span></a></li><li id="n-कार्यशाळा" class="mw-list-item"><a href="/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%B8%E0%A5%8D%E0%A4%AF:%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%B6%E0%A4%BE%E0%A4%B3%E0%A4%BE"><span>कार्यशाळा</span></a></li><li id="n-साहाय्य/मदतकेंद्र" class="mw-list-item"><a href="/wiki/%E0%A4%B8%E0%A4%B9%E0%A4%BE%E0%A4%AF%E0%A5%8D%E0%A4%AF:%E0%A4%86%E0%A4%B6%E0%A4%AF"><span>साहाय्य/मदतकेंद्र</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%85%E0%A4%B2%E0%A5%80%E0%A4%95%E0%A4%A1%E0%A5%80%E0%A4%B2_%E0%A4%AC%E0%A4%A6%E0%A4%B2" title="विकिवरील अलीकडील बदलांची यादी [r]" accesskey="r"><span>अलीकडील बदल</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%A4%E0%A5%87%E0%A4%B9%E0%A5%80" title="कोणतेही पान पाहा [x]" accesskey="x"><span>अविशिष्ट लेख</span></a></li><li id="n-चावडी" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%9A%E0%A4%BE%E0%A4%B5%E0%A4%A1%E0%A5%80"><span>चावडी</span></a></li><li id="n-दूतावास-(Embassy)" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%86%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF_%E0%A4%A6%E0%A5%82%E0%A4%A4%E0%A4%BE%E0%A4%B5%E0%A4%BE%E0%A4%B8"><span>दूतावास (Embassy)</span></a></li><li id="n-ऑनलाइन-शब्दकोश" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%AD%E0%A4%BE%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%B0_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A4%B2%E0%A5%8D%E0%A4%AA/%E0%A4%91%E0%A4%A8%E0%A4%B2%E0%A4%BE%E0%A4%87%E0%A4%A8_%E0%A4%B6%E0%A4%AC%E0%A5%8D%E0%A4%A6%E0%A4%95%E0%A5%8B%E0%A4%B6_%E0%A4%AF%E0%A4%BE%E0%A4%A6%E0%A5%80"><span>ऑनलाइन शब्दकोश</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%E0%A4%AE%E0%A5%81%E0%A4%96%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A0" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="मराठी विकिपीडिया" src="/static/images/mobile/copyright/wikipedia-wordmark-hi.svg" style="width: 7.5em; height: 1.75em;"> <img class="mw-logo-tagline" alt="एक मुक्त ज्ञानकोश" src="/static/images/mobile/copyright/wikipedia-tagline-mr.svg" width="79" height="18" style="width: 4.9375em; height: 1.125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B6%E0%A5%8B%E0%A4%A7%E0%A4%BE" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="विकिपीडिया शोधा [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>शोध</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="शोधा विकिपीडिया" aria-label="शोधा विकिपीडिया" autocapitalize="sentences" title="विकिपीडिया शोधा [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="विशेष:शोधा"> </div> <button class="cdx-button cdx-search-input__end-button">शोधा</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="वैयक्तिक साधने"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_mr.wikipedia.org&uselang=mr" class=""><span>दान</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%A6%E0%A4%B8%E0%A5%8D%E0%A4%AF_%E0%A4%A8%E0%A5%8B%E0%A4%82%E0%A4%A6&returnto=%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="आम्ही आपणास खाते उघडून आल्याची नोंद करण्यास प्रोत्साहीत करत आहोत;अर्थातच, ते नेहमीच आवश्यक नाही." class=""><span>नवीन खाते तयार करा</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%A6%E0%A4%B8%E0%A5%8D%E0%A4%AF_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%B6&returnto=%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="आपणांस सनोंद प्रवेशासाठी प्रोत्साहीत करण्यात येत आहे;अर्थातच, ते अनिवार्य नाही. [o]" accesskey="o" class=""><span>प्रवेश करा(लॉग इन करा)</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="More options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="वैयक्तिक साधने" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">वैयक्तिक साधने</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_mr.wikipedia.org&uselang=mr"><span>दान</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%A6%E0%A4%B8%E0%A5%8D%E0%A4%AF_%E0%A4%A8%E0%A5%8B%E0%A4%82%E0%A4%A6&returnto=%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="आम्ही आपणास खाते उघडून आल्याची नोंद करण्यास प्रोत्साहीत करत आहोत;अर्थातच, ते नेहमीच आवश्यक नाही."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>नवीन खाते तयार करा</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%A6%E0%A4%B8%E0%A5%8D%E0%A4%AF_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%B6&returnto=%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="आपणांस सनोंद प्रवेशासाठी प्रोत्साहीत करण्यात येत आहे;अर्थातच, ते अनिवार्य नाही. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>प्रवेश करा(लॉग इन करा)</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/%E0%A4%B8%E0%A4%B9%E0%A4%BE%E0%A4%AF%E0%A5%8D%E0%A4%AF:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A4%BE%E0%A4%9D%E0%A5%87_%E0%A4%AF%E0%A5%8B%E0%A4%97%E0%A4%A6%E0%A4%BE%E0%A4%A8" title="या अंकपत्त्यावरुन झालेले संपादन [y]" accesskey="y"><span>योगदान</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A4%BE%E0%A4%9D%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%9A%E0%A4%B0%E0%A5%8D%E0%A4%9A%E0%A4%BE" title="या अंकपत्त्यापासून झालेल्या संपादनांबद्दल चर्चा [n]" accesskey="n"><span>चर्चा पान</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">Beginning</div> </a> </li> <li id="toc-गॅमा_व_पाय_फल_(फलन)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#गॅमा_व_पाय_फल_(फलन)"> <div class="vector-toc-text"> <span class="vector-toc-numb">१</span> <span>गॅमा व पाय फल (फलन)</span> </div> </a> <ul id="toc-गॅमा_व_पाय_फल_(फलन)-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">क्रमगुणित</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in ८० languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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">८० languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Fakulteit_(wiskunde)" title="Fakulteit (wiskunde) – अफ्रिकान्स" lang="af" hreflang="af" data-title="Fakulteit (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="अफ्रिकान्स" 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 – अम्हारिक" lang="am" hreflang="am" data-title="ፋክቶሪያል" data-language-autonym="አማርኛ" data-language-local-name="अम्हारिक" 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 – अरबी" lang="ar" hreflang="ar" data-title="عاملي" data-language-autonym="العربية" data-language-local-name="अरबी" 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 – अस्तुरियन" lang="ast" hreflang="ast" data-title="Factorial" data-language-autonym="Asturianu" data-language-local-name="अस्तुरियन" 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 – अझरबैजानी" lang="az" hreflang="az" data-title="Faktorial" data-language-autonym="Azərbaycanca" data-language-local-name="अझरबैजानी" 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 – बष्किर" lang="ba" hreflang="ba" data-title="Факториал" data-language-autonym="Башҡортса" data-language-local-name="बष्किर" 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 – बेलारुशियन" lang="be" hreflang="be" data-title="Фактарыял" data-language-autonym="Беларуская" data-language-local-name="बेलारुशियन" 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 – बल्गेरियन" lang="bg" hreflang="bg" data-title="Факториел" data-language-autonym="Български" data-language-local-name="बल्गेरियन" 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 – बंगाली" lang="bn" hreflang="bn" data-title="গৌণিক" data-language-autonym="বাংলা" data-language-local-name="बंगाली" 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 – बोस्नियन" lang="bs" hreflang="bs" data-title="Faktorijel" data-language-autonym="Bosanski" data-language-local-name="बोस्नियन" 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 – कातालान" lang="ca" hreflang="ca" data-title="Factorial" data-language-autonym="Català" data-language-local-name="कातालान" 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 – मध्य कुर्दिश" lang="ckb" hreflang="ckb" data-title="فاکتۆریێل" data-language-autonym="کوردی" data-language-local-name="मध्य कुर्दिश" 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 – झेक" lang="cs" hreflang="cs" data-title="Faktoriál" data-language-autonym="Čeština" data-language-local-name="झेक" 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 – चूवाश" lang="cv" hreflang="cv" data-title="Факториал" data-language-autonym="Чӑвашла" data-language-local-name="चूवाश" 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) – डॅनिश" lang="da" hreflang="da" data-title="Fakultet (matematik)" data-language-autonym="Dansk" data-language-local-name="डॅनिश" 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) – जर्मन" lang="de" hreflang="de" data-title="Fakultät (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="जर्मन" 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 – ग्रीक" lang="el" hreflang="el" data-title="Παραγοντικό" data-language-autonym="Ελληνικά" data-language-local-name="ग्रीक" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="संपादन"><a href="https://en.wikipedia.org/wiki/Factorial" title="Factorial – इंग्रजी" lang="en" hreflang="en" data-title="Factorial" data-language-autonym="English" data-language-local-name="इंग्रजी" 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 – एस्परान्टो" lang="eo" hreflang="eo" data-title="Faktorialo" data-language-autonym="Esperanto" data-language-local-name="एस्परान्टो" 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 – स्पॅनिश" lang="es" hreflang="es" data-title="Factorial" data-language-autonym="Español" data-language-local-name="स्पॅनिश" 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 – इस्टोनियन" lang="et" hreflang="et" data-title="Faktoriaal" data-language-autonym="Eesti" data-language-local-name="इस्टोनियन" 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 – बास्क" lang="eu" hreflang="eu" data-title="Faktorial" data-language-autonym="Euskara" data-language-local-name="बास्क" 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 – फारसी" lang="fa" hreflang="fa" data-title="فاکتوریل" data-language-autonym="فارسی" data-language-local-name="फारसी" 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 – फिनिश" lang="fi" hreflang="fi" data-title="Kertoma" data-language-autonym="Suomi" data-language-local-name="फिनिश" 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 – फ्रेंच" lang="fr" hreflang="fr" data-title="Factorielle" data-language-autonym="Français" data-language-local-name="फ्रेंच" 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 – गॅलिशियन" lang="gl" hreflang="gl" data-title="Factorial" data-language-autonym="Galego" data-language-local-name="गॅलिशियन" 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) – हिब्रू" lang="he" hreflang="he" data-title="עצרת (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="हिब्रू" 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 – हिंदी" lang="hi" hreflang="hi" data-title="क्रमगुणित" data-language-autonym="हिन्दी" data-language-local-name="हिंदी" 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/Faktorijel" title="Faktorijel – क्रोएशियन" lang="hr" hreflang="hr" data-title="Faktorijel" data-language-autonym="Hrvatski" data-language-local-name="क्रोएशियन" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Faktori%C3%A1lis" title="Faktoriális – हंगेरियन" lang="hu" hreflang="hu" data-title="Faktoriális" data-language-autonym="Magyar" data-language-local-name="हंगेरियन" 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 – आर्मेनियन" lang="hy" hreflang="hy" data-title="Ֆակտորիալ" data-language-autonym="Հայերեն" data-language-local-name="आर्मेनियन" 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 – इंटरलिंग्वा" lang="ia" hreflang="ia" data-title="Factorial" data-language-autonym="Interlingua" data-language-local-name="इंटरलिंग्वा" 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 – इबान" lang="iba" hreflang="iba" data-title="Faktorial" data-language-autonym="Jaku Iban" data-language-local-name="इबान" 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 – इंडोनेशियन" lang="id" hreflang="id" data-title="Faktorial" data-language-autonym="Bahasa Indonesia" data-language-local-name="इंडोनेशियन" 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 – इडौ" lang="io" hreflang="io" data-title="Faktorialo" data-language-autonym="Ido" data-language-local-name="इडौ" 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 – आईसलँडिक" lang="is" hreflang="is" data-title="Aðfeldi" data-language-autonym="Íslenska" data-language-local-name="आईसलँडिक" 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 – इटालियन" lang="it" hreflang="it" data-title="Fattoriale" data-language-autonym="Italiano" data-language-local-name="इटालियन" 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 – जपानी" lang="ja" hreflang="ja" data-title="階乗" data-language-autonym="日本語" data-language-local-name="जपानी" 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 – जॉर्जियन" lang="ka" hreflang="ka" data-title="მათემატიკური ფაქტორიალი" data-language-autonym="ქართული" data-language-local-name="जॉर्जियन" 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 – कझाक" lang="kk" hreflang="kk" data-title="Факториал" data-language-autonym="Қазақша" data-language-local-name="कझाक" 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 – कन्नड" lang="kn" hreflang="kn" data-title="ಕ್ರಮಗುಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="कन्नड" 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) – कोरियन" lang="ko" hreflang="ko" data-title="계승 (수학)" data-language-autonym="한국어" data-language-local-name="कोरियन" 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 – लॅटिन" lang="la" hreflang="la" data-title="Factorialis" data-language-autonym="Latina" data-language-local-name="लॅटिन" 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 – लिथुआनियन" lang="lt" hreflang="lt" data-title="Faktorialas" data-language-autonym="Lietuvių" data-language-local-name="लिथुआनियन" 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 – लात्व्हियन" lang="lv" hreflang="lv" data-title="Faktoriāls" data-language-autonym="Latviešu" data-language-local-name="लात्व्हियन" 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 – मॅसेडोनियन" lang="mk" hreflang="mk" data-title="Факториел" data-language-autonym="Македонски" data-language-local-name="मॅसेडोनियन" 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 – मल्याळम" lang="ml" hreflang="ml" data-title="ക്രമഗുണിതം" data-language-autonym="മലയാളം" data-language-local-name="मल्याळम" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Faktorial" title="Faktorial – मलय" lang="ms" hreflang="ms" data-title="Faktorial" data-language-autonym="Bahasa Melayu" data-language-local-name="मलय" 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) – डच" lang="nl" hreflang="nl" data-title="Faculteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="डच" 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 – नॉर्वेजियन न्योर्स्क" lang="nn" hreflang="nn" data-title="Fakultet i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="नॉर्वेजियन न्योर्स्क" 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) – नॉर्वेजियन बोकमाल" lang="nb" hreflang="nb" data-title="Fakultet (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="नॉर्वेजियन बोकमाल" 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 – ऑक्सितान" lang="oc" hreflang="oc" data-title="Factoriala" data-language-autonym="Occitan" data-language-local-name="ऑक्सितान" 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 – पंजाबी" lang="pa" hreflang="pa" data-title="ਕ੍ਰਮਗੁਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="पंजाबी" 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 – पोलिश" lang="pl" hreflang="pl" data-title="Silnia" data-language-autonym="Polski" data-language-local-name="पोलिश" 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 – पोर्तुगीज" lang="pt" hreflang="pt" data-title="Fatorial" data-language-autonym="Português" data-language-local-name="पोर्तुगीज" 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 – रोमानियन" lang="ro" hreflang="ro" data-title="Factorial" data-language-autonym="Română" data-language-local-name="रोमानियन" 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 – रशियन" lang="ru" hreflang="ru" data-title="Факториал" data-language-autonym="Русский" data-language-local-name="रशियन" 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 – सिसिलियन" lang="scn" hreflang="scn" data-title="Fatturiali" data-language-autonym="Sicilianu" data-language-local-name="सिसिलियन" 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 – सर्बो-क्रोएशियन" lang="sh" hreflang="sh" data-title="Faktorijel" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="सर्बो-क्रोएशियन" 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 – सिंहला" lang="si" hreflang="si" data-title="ක්‍රමාරෝපිතය" data-language-autonym="සිංහල" data-language-local-name="सिंहला" 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 – स्लोव्हाक" lang="sk" hreflang="sk" data-title="Faktoriál" data-language-autonym="Slovenčina" data-language-local-name="स्लोव्हाक" 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) – स्लोव्हेनियन" lang="sl" hreflang="sl" data-title="Fakulteta (funkcija)" data-language-autonym="Slovenščina" data-language-local-name="स्लोव्हेनियन" 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 – अल्बानियन" lang="sq" hreflang="sq" data-title="Faktoriali" data-language-autonym="Shqip" data-language-local-name="अल्बानियन" 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 – सर्बियन" lang="sr" hreflang="sr" data-title="Факторијел" data-language-autonym="Српски / srpski" data-language-local-name="सर्बियन" 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) – स्वीडिश" lang="sv" hreflang="sv" data-title="Fakultet (matematik)" data-language-autonym="Svenska" data-language-local-name="स्वीडिश" 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 – तामिळ" lang="ta" hreflang="ta" data-title="தொடர் பெருக்கம்" data-language-autonym="தமிழ்" data-language-local-name="तामिळ" 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 – थाई" lang="th" hreflang="th" data-title="แฟกทอเรียล" data-language-autonym="ไทย" data-language-local-name="थाई" 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 – टागालोग" lang="tl" hreflang="tl" data-title="Paktoryal" data-language-autonym="Tagalog" data-language-local-name="टागालोग" 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 – तुर्की" lang="tr" hreflang="tr" data-title="Faktöriyel" data-language-autonym="Türkçe" data-language-local-name="तुर्की" 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 – युक्रेनियन" lang="uk" hreflang="uk" data-title="Факторіал" data-language-autonym="Українська" data-language-local-name="युक्रेनियन" 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 – उर्दू" lang="ur" hreflang="ur" data-title="عاملیہ" data-language-autonym="اردو" data-language-local-name="उर्दू" 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 – उझ्बेक" lang="uz" hreflang="uz" data-title="Faktorial" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="उझ्बेक" 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 – व्हिएतनामी" lang="vi" hreflang="vi" data-title="Giai thừa" data-language-autonym="Tiếng Việt" data-language-local-name="व्हिएतनामी" 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 – व्हू चिनी" lang="wuu" hreflang="wuu" data-title="阶乘" data-language-autonym="吴语" data-language-local-name="व्हू चिनी" 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 – चीनी" lang="zh" hreflang="zh" data-title="階乘" data-language-autonym="中文" data-language-local-name="चीनी" 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 – कँटोनीज" lang="yue" hreflang="yue" data-title="階乘" data-language-autonym="粵語" data-language-local-name="कँटोनीज" 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="आंतरभाषिक दुवे संपादा" class="wbc-editpage">दुवे संपादा</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="नामविश्वे"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/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="आशयाचे पान पहा [c]" accesskey="c"><span>लेख</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E0%A4%9A%E0%A4%B0%E0%A5%8D%E0%A4%9A%E0%A4%BE:%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&action=edit&redlink=1" rel="discussion" class="new" title="आशय पानाबद्दलची चर्चा (पान अस्तित्वात नाही) [t]" accesskey="t"><span>चर्चा</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">मराठी</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="दृष्ये"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/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"><span>वाचा</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%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&action=edit" title="लेखपान मूळस्वरूपात संपादीत करा,तेथे मराठी टायपींगसाठी युएलएस कार्यान्वित करू शकता. [e]" accesskey="e"><span>संपादन</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%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&action=history" title="या पानाच्या जुन्या आवृत्या. [h]" accesskey="h"><span>इतिहास पहा</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="साधनपेटी" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">साधनपेटी</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/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"><span>वाचा</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%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&action=edit" title="लेखपान मूळस्वरूपात संपादीत करा,तेथे मराठी टायपींगसाठी युएलएस कार्यान्वित करू शकता. [e]" accesskey="e"><span>संपादन</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%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&action=history"><span>इतिहास पहा</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AF%E0%A5%87%E0%A4%A5%E0%A5%87_%E0%A4%95%E0%A4%BE%E0%A4%AF_%E0%A4%9C%E0%A5%8B%E0%A4%A1%E0%A4%B2%E0%A5%87_%E0%A4%86%E0%A4%B9%E0%A5%87/%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="येथे जोडलेल्या सर्व विकिपानांची यादी [j]" accesskey="j"><span>येथे काय जोडले आहे</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%BE%E0%A4%82%E0%A4%A7%E0%A4%B2%E0%A5%87%E0%A4%B2%E0%A5%87%E0%A4%85%E0%A4%B2%E0%A4%BF%E0%A4%95%E0%A4%A1%E0%A5%80%E0%A4%B2%E0%A4%AC%E0%A4%A6%E0%A4%B2/%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" rel="nofollow" title="येथुन जोडलेल्या सर्व पानांवरील अलीकडील बदल [k]" accesskey="k"><span>या पृष्ठासंबंधीचे बदल</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=mr" title="चित्रे किंवा माध्यम संचिका चढवा [u]" accesskey="u"><span>संचिका चढवा</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7_%E0%A4%AA%E0%A4%BE%E0%A4%A8%E0%A5%87" title="सर्व विशेष पृष्ठांची यादी [q]" accesskey="q"><span>विशेष पृष्ठे</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%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&oldid=2176557" title="पानाच्या या आवर्तनाचा शाश्वत दुवा"><span>शाश्वत दुवा</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%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&action=info" title="या पानाबाबत अधिक माहिती"><span>पानाबद्दलची माहिती</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B8%E0%A4%82%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%AD%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A4%BE&page=%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&id=2176557&wpFormIdentifier=titleform" title="हे पृष्ठ उद्धृत कसे करावे याची माहिती"><span>लेखाचा संदर्भ द्या</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:UrlShortener&url=https%3A%2F%2Fmr.wikipedia.org%2Fwiki%2F%25E0%25A4%2595%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A4%25AE%25E0%25A4%2597%25E0%25A5%2581%25E0%25A4%25A3%25E0%25A4%25BF%25E0%25A4%25A4"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:QrCode&url=https%3A%2F%2Fmr.wikipedia.org%2Fwiki%2F%25E0%25A4%2595%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A4%25AE%25E0%25A4%2597%25E0%25A5%2581%25E0%25A4%25A3%25E0%25A4%25BF%25E0%25A4%25A4"><span>Download QR code</span></a></li><li id="t-shorturl" class="mw-list-item"><a href="//mr.wikipedia.org/s/5i34" title="Copy this short link for sharing"><span>लघु यूआरएल(Short URL)</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> छापा/ निर्यात करा </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AA%E0%A5%81%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%95&bookcmd=book_creator&referer=%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"><span>ग्रंथ तयार करा</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:DownloadAsPdf&page=%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&action=show-download-screen"><span>PDF म्हणून उतरवा</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%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&printable=yes" title="या पानाची छापण्यायोग्य आवृत्ती [p]" accesskey="p"><span>छापण्यायोग्य आवृत्ती</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> इतर प्रकल्पात </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Factorial_(function)" hreflang="en"><span>विकिमिडिया कॉमन्स</span></a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z13667" hreflang="en"><span>Wikifunctions</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q120976" title="माहिती भांडारात जोडलेल्या कलमाचा दुवा [g]" accesskey="g"><span>विकिडाटा कलम</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">विकिपीडिया, मुक्‍त ज्ञानकोशातून</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="mr" dir="ltr"><table class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"> <caption>काही निवडक संख्यांचे क्रमगुणीत; काही संख्या अचूकपणे प्रदर्शित करण्यासाठी घातांकाच्या रूपात लिहिल्या आहेत. </caption> <tbody><tr> <th><b>n</b> </th> <th><b>n!</b> </th></tr> <tr> <td>०</td> <td>१ </td></tr> <tr> <td>१</td> <td>१ </td></tr> <tr> <td>२</td> <td>२ </td></tr> <tr> <td>३</td> <td>३ </td></tr> <tr> <td>४</td> <td>२४ </td></tr> <tr> <td>५</td> <td><span style="white-space:nowrap">१२०</span> </td></tr> <tr> <td>६</td> <td><span style="white-space:nowrap">७२०</span> </td></tr> <tr> <td>७</td> <td><span style="white-space:nowrap">५,०४०</span> </td></tr> <tr> <td>८</td> <td><span style="white-space:nowrap">४०,३२०</span> </td></tr> <tr> <td><span style="white-space:nowrap">९</span></td> <td><span style="white-space:nowrap">३,६२,८८०</span> </td></tr> <tr> <td><span style="white-space:nowrap">१०</span></td> <td><span style="white-space:nowrap">३६,२८,८००</span> </td></tr> <tr> <td><span style="white-space:nowrap">११</span></td> <td><span style="white-space:nowrap">३,९९,१६,८००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१२</span></td> <td><span style="white-space:nowrap">४७,९०,०१,६००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१३</span></td> <td><span style="white-space:nowrap">६,२२,७०,२०,८००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१४</span></td> <td><span style="white-space:nowrap">८७,१७,८२,९१,२००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१५</span></td> <td><span style="white-space:nowrap">१३,०७,६७,४३,६८,०००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१६</span></td> <td><span style="white-space:nowrap">२,०९,२२,७८,९८,८८,०००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१७</span></td> <td><span style="white-space:nowrap">३५,५६,८७,४२,८०,९६,०००</span> </td></tr> <tr> <td><span style="white-space:nowrap">१८</span></td> <td><span style="white-space:nowrap">६,४०,२३,७३,७०,५७,२८,०००.</span> </td></tr> <tr> <td><span style="white-space:nowrap">१९</span></td> <td><span style="white-space:nowrap">१,२१,६४,५१,००,४०,८८,३२,०००.</span> </td></tr> <tr> <td><span style="white-space:nowrap">२०</span></td> <td><span style="white-space:nowrap">२४,३२,९०,२०,०८,१७,६६,४०,०००.</span> </td></tr> <tr> <td><span style="white-space:nowrap">२५</span> </td> <td style="text-align:left"><span style="white-space:nowrap">१.५५१<span style="margin-left:०.२५em">१</span>२१<span style="margin-left:०.२५em">०</span>०४<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>२५</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">५०</span> </td> <td style="text-align:left"><span style="white-space:nowrap">३.०४१<span style="margin-left:०.२५em">४</span>०९<span style="margin-left:०.२५em">३</span>२०<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>६४</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">७०</span> </td> <td style="text-align:left"><span style="white-space:nowrap">१.१९७<span style="margin-left:०.२५em">८</span>५७<span style="margin-left:०.२५em">१</span>६७<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१००</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">१००</span> </td> <td style="text-align:left"><span style="white-space:nowrap">९.३३२<span style="margin-left:०.२५em">६</span>२१<span style="margin-left:०.२५em">५</span>४४<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१५७</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">४५०</span> </td> <td style="text-align:left"><span style="white-space:nowrap">१.७३३<span style="margin-left:०.२५em">३</span>६८<span style="margin-left:०.२५em">७</span>३३<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१,०००</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">१,०००</span> </td> <td style="text-align:left"><span style="white-space:nowrap">४.०२३<span style="margin-left:०.२५em">८</span>७२<span style="margin-left:०.२५em">६</span>०१<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>२,५६७</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">३,२४९</span> </td> <td style="text-align:left"><span style="white-space:nowrap">६.४१२<span style="margin-left:०.२५em">३</span>३७<span style="margin-left:०.२५em">६</span>८८<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१०,०००</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">१०,०००</span> </td> <td style="text-align:left"><span style="white-space:nowrap">२.८४६<span style="margin-left:०.२५em">२</span>५९<span style="margin-left:०.२५em">६</span>८१<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>३५,६५९</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">२५,२०६</span> </td> <td style="text-align:left"><span style="white-space:nowrap">१.२०५<span style="margin-left:०.२५em">७</span>०३<span style="margin-left:०.२५em">४</span>३८<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१,००,०००</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">१,००,०००</span> </td> <td style="text-align:left"><span style="white-space:nowrap">२.८२४<span style="margin-left:०.२५em">२</span>२९<span style="margin-left:०.२५em">४</span>०८<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>४,५६,५७३</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">२,०५,०२३</span> </td> <td style="text-align:left"><span style="white-space:nowrap">२.५०३<span style="margin-left:०.२५em">८</span>९८<span style="margin-left:०.२५em">९</span>३२<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>१०,००,००४</sup></span> </td></tr> <tr> <td><span style="white-space:nowrap">१०,००,०००</span> </td> <td style="text-align:left"><span style="white-space:nowrap">८.२६३<span style="margin-left:०.२५em">९</span>३१<span style="margin-left:०.२५em">६</span>८८<span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<sup>५५,६५,७०८</sup></span> </td></tr></tbody></table> <p>गणितामध्ये कोणत्याही ऋण नसलेल्या पूर्णांकाचा nचा <b>क्रमगुणित (factorial- फॅक्टोरियल)</b> हा n! ने दर्शवतात.<br />क्रमगुणित म्हणजे ती संख्या व तिच्यापेक्षा लहान धन पूर्णांकाचा गुणाकार होय. </p> <pre> <b>n! = n . (n-१) . (n-२) . (n-३) . (n-४) ..... ४ . ३ . २ . १</b> </pre> <p>उदाहरणार्थ, </p> <pre> <b><span style="white-space:nowrap">५</span>! = <span style="white-space:nowrap">५</span> . <span style="white-space:nowrap">४</span> . <span style="white-space:nowrap">३</span> . <span style="white-space:nowrap">२</span> . <span style="white-space:nowrap">१</span> = <span style="white-space:nowrap">१२०</span></b> </pre> <p>याचा उपयोग बीजगणित, मांडणी व जुळवणी मध्ये केला जातो. </p><p><span style="white-space:nowrap">०</span>!चे मूल्य! <span style="white-space:nowrap">१</span> आहे. कारण, </p> <pre> ⇒n!=n×(n−<span style="white-space:nowrap">१</span>)! समजा nची किंमत १ धरल्यास, ⇒<span style="white-space:nowrap">१</span>!=<span style="white-space:nowrap">१</span>!×(<span style="white-space:nowrap">१</span>−<span style="white-space:nowrap">१</span>)! ⇒<span style="white-space:nowrap">१</span>!=<span style="white-space:nowrap">१</span>!×(<span style="white-space:nowrap">०</span>)! </pre> <p>डावी बाजू = उजवी बाजू असले पाहिजे, हा नियम पूर्ण होण्यासाठी, <span style="white-space:nowrap">०</span>! = <span style="white-space:nowrap">१</span> असणे हे गरजेचे आहे. </p> <div class="mw-heading mw-heading2"><h2 id="गॅमा_व_पाय_फल_(फलन)"><span id=".E0.A4.97.E0.A5.85.E0.A4.AE.E0.A4.BE_.E0.A4.B5_.E0.A4.AA.E0.A4.BE.E0.A4.AF_.E0.A4.AB.E0.A4.B2_.28.E0.A4.AB.E0.A4.B2.E0.A4.A8.29"></span>गॅमा व पाय फल (फलन)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&action=edit&section=1" title="विभाग: गॅमा व पाय फल (फलन) संपादा"><span>संपादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="noprint relarticle mainarticle"><i>मुख्य पान: <a href="/wiki/%E0%A4%97%E0%A5%85%E0%A4%AE%E0%A4%BE_%E0%A4%AB%E0%A4%B2" title="गॅमा फल">गॅमा फल</a></i></div></dd></dl> <p>गैर-ऋण (ऋण नसलेला) पूर्णांकाशिवाय <a href="/wiki/%E0%A4%85%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अपूर्णांक संख्या">अपूर्णांकांचा</a> सुद्धा क्रमगुणित काढला जाऊ शकतो, परंतु यासाठी गणितीय विश्लेषणातील अधिक प्रगत साधने (क्रिया) आवश्यक आहेत. </p><p>गॅमा फल हे फॅक्टोरियलचे विस्तारित रूप आहे. आणि हे <span class="texhtml">Γ(<i>z</i>)</span> ने दर्शवतात. गॅमा फलाचे उत्तर हे क्रमगुणित मूल्यांमध्ये येते फक्त गॅमा फलातील चलाची किंमत ही १ ने बदलल्यास म्हणजे <span class="texhtml">Γ(<i>2</i>)</span> = 1!,<span class="texhtml">Γ(<i>3</i>)</span> = 2!, नॉन-पॉझिटिव्ह पूर्णांक वगळता सर्व <a href="/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संमिश्र संख्या">संमिश्र संख्यां</a>साठी ते परिभाषित केले जाते, आणि जर वास्तविक भाग हा धन असेल तर,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9699b7cf771997e036ad88f17f835e96f3f959" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.395ex; height:5.843ex;" alt="{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.}"></span>प्रत्येक गैर-ऋण <a href="/wiki/%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="पूर्ण संख्या">पूर्णांक संख्या</a> n साठी क्रमगुणित आणि गॅमा फल यांमधील संबंध हा <span class="texhtml"><i>n</i>! = Γ(<i>n</i> + 1)</span> असा असतो. </p><p><br /> यूलरचे गॅमा फलाचे मूळचे सूत्र <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d95a36afd634d090227d1cc74411af1236cd09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:23.51ex; height:10.343ex;" alt="{\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}.}"></span> </p><p><br /> <a href="/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%B2_%E0%A4%AB%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%A6%E0%A4%B0%E0%A4%BF%E0%A4%B6_%E0%A4%97%E0%A4%BE%E0%A4%89%E0%A4%B8" title="कार्ल फ्रीदरिश गाउस">कार्ल फ्रेडरिक गॉस</a> यांनी समान हेच दर्शविण्यासाठी <span class="texhtml">Π(<i>z</i>)</span> चिन्ह वापरले, परंतु चलाची किंमत ही १ ने बदलली आहे, जेणेकरून ते गैर-ऋणात्मक पूर्णांकांच्या क्रमगुणितांसाठी बरोबर ठरेन. पाय <a href="/wiki/%E0%A4%AB%E0%A4%B2_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="फल (गणित)">फलाची</a> व्याख्या<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (z)=\int _{0}^{\infty }t^{z}e^{-t}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (z)=\int _{0}^{\infty }t^{z}e^{-t}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb26c7e7ba7669dc4eaf14851318cd586b552cc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.584ex; height:5.843ex;" alt="{\displaystyle \Pi (z)=\int _{0}^{\infty }t^{z}e^{-t}\,dt.}"></span>तच </p><p><br /> पाय आणि गॅमा फलामधील संबंधाचे सूत्र <span class="texhtml">Π(<i>z</i>) = Γ(<i>z</i> + 1)</span>. </p><p>अशाप्रकारे प्रत्येक गैर-ऋण पूर्णांक संख्या n साठी, <span class="texhtml">Π(<i>n</i>) = <i>n</i>!</span> . </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Generalized_factorial_function.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Generalized_factorial_function.svg/350px-Generalized_factorial_function.svg.png" decoding="async" width="350" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Generalized_factorial_function.svg/525px-Generalized_factorial_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Generalized_factorial_function.svg/700px-Generalized_factorial_function.svg.png 2x" data-file-width="500" data-file-height="315" /></a><figcaption>फॅक्टोरियल फंक्शन (क्रमगुणित फल), ऋण पूर्णांक वगळता सर्व वास्तविक संख्यांसाठी सामान्यीकृत. उदाहरणार्थ, <span class="nowrap">0! = 1! = 1</span>, <span class="nowrap">(−1/2)! = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><span class="texhtml">π</span></span></span></span>, <span class="nowrap">1/2! = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><span class="texhtml">π</span></span></span>/2</span>.</figcaption></figure> <p>या व्यतिरिक्त , पाय फल हे क्रमगुणिताप्रमाणे परिभाषित केलेल्या प्रत्येक <span class="texhtml mvar" style="font-style:italic;">z</span> (परिभाषित केलेली <a href="/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संमिश्र संख्या">संमिश्र संख्यां</a>) ला पुनरावृत्ती दर्शविते. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (z)=z\Pi (z-1)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (z)=z\Pi (z-1)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fbf125086c6e4ed0dc737cfc1ff8ddfb11ae67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.504ex; height:2.843ex;" alt="{\displaystyle \Pi (z)=z\Pi (z-1)\,.}"></span> </p><p><br /> हे पुनरावृत्ती संबंध आहे. याला गॅमा फलात लिहील्यास, हे पुनरावृत्त राहत नाही.<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (n+1)=n\Gamma (n)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (n+1)=n\Gamma (n)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529e5a287b8a8fcfb021de7f96eaacdfa586abf0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.844ex; height:2.843ex;" alt="{\displaystyle \Gamma (n+1)=n\Gamma (n)\,.}"></span> </p><p><br /> अर्ध-पूर्णांक संख्यांना या फलांची किंमत ही त्यांच्यापैकी एकाद्वारे निर्धारित केली जाते:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left({\frac {1}{2}}\right)=\left(-{\frac {1}{2}}\right)!=\Pi \left(-{\frac {1}{2}}\right)={\sqrt {\pi }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left({\frac {1}{2}}\right)=\left(-{\frac {1}{2}}\right)!=\Pi \left(-{\frac {1}{2}}\right)={\sqrt {\pi }}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b6395521650b9b935230fe93f665bf57a3cec1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.09ex; height:6.176ex;" alt="{\displaystyle \Gamma \left({\frac {1}{2}}\right)=\left(-{\frac {1}{2}}\right)!=\Pi \left(-{\frac {1}{2}}\right)={\sqrt {\pi }}\,,}"></span> हे <span class="texhtml"><i>n</i> ∈ <b>N</b></span> साठी,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\Gamma \left({\frac {1}{2}}+n\right)=\left(-{\frac {1}{2}}+n\right)!=\Pi \left(-{\frac {1}{2}}+n\right)\\[5pt]={}&{\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2k-1}{2}}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}={\frac {(2n-1)!}{2^{2n-1}(n-1)!}}{\sqrt {\pi }}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\Gamma \left({\frac {1}{2}}+n\right)=\left(-{\frac {1}{2}}+n\right)!=\Pi \left(-{\frac {1}{2}}+n\right)\\[5pt]={}&{\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2k-1}{2}}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}={\frac {(2n-1)!}{2^{2n-1}(n-1)!}}{\sqrt {\pi }}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/073bf6ce7a3fbee30bd905a7c75e13b7ce9a80df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:51.038ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}&\Gamma \left({\frac {1}{2}}+n\right)=\left(-{\frac {1}{2}}+n\right)!=\Pi \left(-{\frac {1}{2}}+n\right)\\[5pt]={}&{\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2k-1}{2}}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}={\frac {(2n-1)!}{2^{2n-1}(n-1)!}}{\sqrt {\pi }}\,.\end{aligned}}}"></span>उदाहरणार्थ, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left({\frac {9}{2}}\right)={\frac {7}{2}}!=\Pi \left({\frac {7}{2}}\right)={\frac {1}{2}}\cdot {\frac {3}{2}}\cdot {\frac {5}{2}}\cdot {\frac {7}{2}}{\sqrt {\pi }}={\frac {8!}{4^{4}4!}}{\sqrt {\pi }}={\frac {7!}{2^{7}3!}}{\sqrt {\pi }}={\frac {105}{16}}{\sqrt {\pi }}\approx 11.631\,728\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>!</mo> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>105</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>≈</mo> <mn>11.631</mn> <mspace width="thinmathspace" /> <mn>728</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left({\frac {9}{2}}\right)={\frac {7}{2}}!=\Pi \left({\frac {7}{2}}\right)={\frac {1}{2}}\cdot {\frac {3}{2}}\cdot {\frac {5}{2}}\cdot {\frac {7}{2}}{\sqrt {\pi }}={\frac {8!}{4^{4}4!}}{\sqrt {\pi }}={\frac {7!}{2^{7}3!}}{\sqrt {\pi }}={\frac {105}{16}}{\sqrt {\pi }}\approx 11.631\,728\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bda7e6192a3d5ed13e6ef8da06501973b35ca7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:92.739ex; height:6.176ex;" alt="{\displaystyle \Gamma \left({\frac {9}{2}}\right)={\frac {7}{2}}!=\Pi \left({\frac {7}{2}}\right)={\frac {1}{2}}\cdot {\frac {3}{2}}\cdot {\frac {5}{2}}\cdot {\frac {7}{2}}{\sqrt {\pi }}={\frac {8!}{4^{4}4!}}{\sqrt {\pi }}={\frac {7!}{2^{7}3!}}{\sqrt {\pi }}={\frac {105}{16}}{\sqrt {\pi }}\approx 11.631\,728\ldots }"></span> </p><p>सर्व <span class="texhtml"><i>n</i> ∈ <b>N</b></span> साठी, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left({\frac {1}{2}}-n\right)=\left(-{\frac {1}{2}}-n\right)!=\Pi \left(-{\frac {1}{2}}-n\right)={\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2}{1-2k}}={\frac {\left(-4\right)^{n}n!}{(2n)!}}{\sqrt {\pi }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left({\frac {1}{2}}-n\right)=\left(-{\frac {1}{2}}-n\right)!=\Pi \left(-{\frac {1}{2}}-n\right)={\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2}{1-2k}}={\frac {\left(-4\right)^{n}n!}{(2n)!}}{\sqrt {\pi }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f95bad77594f19043cb753b1e58e566407797a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.994ex; height:6.843ex;" alt="{\displaystyle \Gamma \left({\frac {1}{2}}-n\right)=\left(-{\frac {1}{2}}-n\right)!=\Pi \left(-{\frac {1}{2}}-n\right)={\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2}{1-2k}}={\frac {\left(-4\right)^{n}n!}{(2n)!}}{\sqrt {\pi }}\,.}"></span> उदाहरणार्थ, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left(-{\frac {5}{2}}\right)=\left(-{\frac {7}{2}}\right)!=\Pi \left(-{\frac {7}{2}}\right)={\frac {2}{-1}}\cdot {\frac {2}{-3}}\cdot {\frac {2}{-5}}{\sqrt {\pi }}={\frac {\left(-4\right)^{3}3!}{6!}}{\sqrt {\pi }}=-{\frac {8}{15}}{\sqrt {\pi }}\approx -0.945\,308\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> <mo>!</mo> </mrow> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>15</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>≈</mo> <mo>−</mo> <mn>0.945</mn> <mspace width="thinmathspace" /> <mn>308</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left(-{\frac {5}{2}}\right)=\left(-{\frac {7}{2}}\right)!=\Pi \left(-{\frac {7}{2}}\right)={\frac {2}{-1}}\cdot {\frac {2}{-3}}\cdot {\frac {2}{-5}}{\sqrt {\pi }}={\frac {\left(-4\right)^{3}3!}{6!}}{\sqrt {\pi }}=-{\frac {8}{15}}{\sqrt {\pi }}\approx -0.945\,308\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1446116e6bd8c44f61cae9fe36a661691d127382" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:97.012ex; height:6.676ex;" alt="{\displaystyle \Gamma \left(-{\frac {5}{2}}\right)=\left(-{\frac {7}{2}}\right)!=\Pi \left(-{\frac {7}{2}}\right)={\frac {2}{-1}}\cdot {\frac {2}{-3}}\cdot {\frac {2}{-5}}{\sqrt {\pi }}={\frac {\left(-4\right)^{3}3!}{6!}}{\sqrt {\pi }}=-{\frac {8}{15}}{\sqrt {\pi }}\approx -0.945\,308\ldots }"></span> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">"<a dir="ltr" href="https://mr.wikipedia.org/w/index.php?title=क्रमगुणित&oldid=2176557">https://mr.wikipedia.org/w/index.php?title=क्रमगुणित&oldid=2176557</a>" पासून हुडकले</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97" title="विशेष:वर्ग">वर्ग</a>: <ul><li><a href="/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97:%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="वर्ग:गणित">गणित</a></li><li><a href="/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97:%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="वर्ग:अंकगणित">अंकगणित</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">लपविलेला वर्ग: <ul><li><a href="/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97:Pages_with_non-numeric_formatnum_arguments" title="वर्ग:Pages with non-numeric formatnum arguments">Pages with non-numeric formatnum arguments</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"> या पानातील शेवटचा बदल १९ सप्टेंबर २०२२ रोजी १४:३७ वाजता केला गेला.</li> <li id="footer-info-copyright">येथील मजकूर हा <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">क्रियेटीव्ह कॉमन्स अट्रीब्युशन-शेअरअलाईक लायसन्स</a> अंतर्गत उपलब्ध आहे; अतिरिक्त अटी लागू असू शकतात. हे संकेतस्थळ वापरुन आपण याच्या <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">वापरण्याच्या अटी</a> आणि <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">गोपनीयता धोरणांचे</a> पालन करण्यास आपली सहमती देत आहात. Wikipedia® हे <a rel="nofollow" class="external text" href="https://www.wikimediafoundation.org/">Wikimedia Foundation, Inc.</a> एक ना-नफा ना-तोटा संस्थेचे नोंदणीकृत ट्रेडमार्क आहे.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">गुप्तता नीती</a></li> <li id="footer-places-about"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%AE%E0%A4%BE%E0%A4%B9%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A0">विकिपीडिया बद्दल</a></li> <li id="footer-places-disclaimers"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%BE:%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%B8%E0%A4%BE%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%A3_%E0%A4%89%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A4%A6%E0%A4%BE%E0%A4%AF%E0%A4%95%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%B8_%E0%A4%A8%E0%A4%95%E0%A4%BE%E0%A4%B0">उत्तरदायित्वास नकार</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">विकसक</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/mr.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">कुकिंचा तक्ता</a></li> <li id="footer-places-mobileview"><a href="//mr.m.wikipedia.org/w/index.php?title=%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&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">भ्रमणध्वनी-दृश्य</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-gdkxt","wgBackendResponseTime":199,"wgPageParseReport":{"limitreport":{"cputime":"0.183","walltime":"0.263","ppvisitednodes":{"value":8017,"limit":1000000},"postexpandincludesize":{"value":70466,"limit":2097152},"templateargumentsize":{"value":7785,"limit":2097152},"expansiondepth":{"value":29,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":360,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 177.346 1 -total"," 80.96% 143.579 71 साचा:Val"," 36.17% 64.146 83 साचा:Val/delimitnum"," 25.69% 45.558 83 साचा:Val/delimitnum/logic"," 8.17% 14.481 15 साचा:Val/delimitnum/fraction"," 7.31% 12.955 12 साचा:Su"," 5.43% 9.634 3 साचा:Nowrap"," 4.46% 7.916 1 साचा:Main"," 4.38% 7.769 9 साचा:Math"," 3.24% 5.741 2 साचा:Sqrt"]},"cachereport":{"origin":"mw-web.eqiad.main-85db4f49b5-r7f6v","timestamp":"20241114003241","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u0915\u094d\u0930\u092e\u0917\u0941\u0923\u093f\u0924","url":"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","sameAs":"http:\/\/www.wikidata.org\/entity\/Q120976","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q120976","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2021-12-10T11:03:19Z","dateModified":"2022-09-19T09:07:11Z"}</script> </body> </html>