{"id":10756,"date":"2025-05-28T08:40:36","date_gmt":"2025-05-28T06:40:36","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=10756"},"modified":"2026-09-15T14:21:49","modified_gmt":"2026-09-15T12:21:49","slug":"plusieurs-facons-de-calculer-une-factorielle-en-python","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/plusieurs-facons-de-calculer-une-factorielle-en-python\/","title":{"rendered":"Plusieurs fa\u00e7ons de calculer une factorielle en Python"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Nous allons voir sur cette page plusieurs mani\u00e8res de calculer une&nbsp;<em><strong>factorielle en Python<\/strong><\/em>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"188\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle-300x188.png\" alt=\"python factorielle\" class=\"wp-image-3098\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle-300x188.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle-600x375.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle-1024x640.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle-768x480.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-factorielle.png 1080w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\">Qu&rsquo;est-ce qu&rsquo;une factorielle du point de vue math\u00e9matique ?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Du point de vue math\u00e9matiques, la&nbsp;<em>factorielle<\/em>&nbsp;d\u2019un entier naturel&nbsp;<em>n<\/em>&nbsp;est le produit not\u00e9&nbsp;<em>n<\/em>! tel que : <\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"> n!=1\u00d72\u00d73\u00d74\u00d7\u22ef\u00d7(n\u22121)\u00d7n.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5! = 1\u00d72\u00d73\u00d74\u00d75 = 120.<\/li>\n\n\n\n<li>2! = 1\u00d72 = 2.<\/li>\n\n\n\n<li>1! = 1.<\/li>\n\n\n\n<li>0! = 1&nbsp;(par convention).<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Python et factorielle: une approche na\u00efve<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 partir de la d\u00e9finition math\u00e9matique pr\u00e9c\u00e9dente, on peut imaginer une premi\u00e8re approche d\u2019un programme Python nous permettant de calculer une factorielle :<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def factorielle(n):\n   if n == 0:\n      return 1\n   else:\n      F = 1\n      for k in range(2,n+1):\n         F = F * k\n\n      return F<\/pre>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; factorielle(5) \n120<\/code><\/pre>\n<\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Ex\u00e9cution pas \u00e0 pas du programme<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Que s\u2019est-il pass\u00e9 ici pour voir ce r\u00e9sultat ?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>On appelle&nbsp;<em>factorielle(5)<\/em>&nbsp;:<\/li>\n\n\n\n<li>quand on entre dans la fonction&nbsp;<em>factorielle<\/em>, on teste avant tout si l\u2019argument vaut 0, ce qui n\u2019est pas le cas donc on passe \u00e0 la ligne 5.<\/li>\n\n\n\n<li>On initialise alors une variable F \u00e0 1,<\/li>\n\n\n\n<li>puis on entre dans une boucle o\u00f9 la variable&nbsp;<em>k<\/em>&nbsp;varie de 2 \u00e0&nbsp;<em>n<\/em>.<\/li>\n\n\n\n<li>\u00c0 chaque passage, la valeur contenue dans F est multipli\u00e9e par&nbsp;<em>k<\/em>, ce qui donne (sous forme de tableau):<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\" style=\"border-width:1px\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><em>k<\/em><\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">F<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1\u00d72 = 2<\/td><td class=\"has-text-align-center\" data-align=\"center\">2\u00d73 = 6<\/td><td class=\"has-text-align-center\" data-align=\"center\">6\u00d74 = 24<\/td><td class=\"has-text-align-center\" data-align=\"center\">24\u00d75 = 120<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Le r\u00e9sultat final retourn\u00e9 est donc 120.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Python et factorielle: une approche r\u00e9cursive<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On peut remarquer que si on pose \\( f(n)=n! \\)\u00a0alors, $$\\begin{align}f(n+1)&amp;=(n+1)!\\\\&amp;=1\\times 2\\times \\cdots \\times n\\times(n+1)\\\\ &amp; = n! \\times (n+1)\\\\&amp; = f(n)\\times (n+1).\\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut alors imaginer un deuxi\u00e8me programme l\u00e9g\u00e8rement diff\u00e9rent du premier:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def factorielle(n):\n    if n == 0:\n        return 1\n    else:\n        return n  * factorielle(n-1)<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">C\u2019est ce que l\u2019on appelle la&nbsp;<em>forme r\u00e9cursive<\/em>&nbsp;du programme. On l\u2019appelle ainsi car pour calculer la factorielle d\u2019un entier&nbsp;<em>n<\/em>, on fait appel \u00e0 la factorielle de l\u2019entier pr\u00e9c\u00e9dent, \u00e0 l\u2019instar d\u2019une suite r\u00e9cursive de la forme&nbsp;\\( u_{n+1} = f(u_n) \\).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Une variante avec la fonction lambda<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On peut l\u00e9g\u00e8rement simplifier la syntaxe pr\u00e9c\u00e9dente tout en gardant la forme r\u00e9cursive \u00e0 l\u2019aide de la&nbsp;<strong><em>fonction lambda<\/em><\/strong>&nbsp;de Python:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">factorielle = lambda n: n * factorielle(n-1) if n != 0 else 1\n<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Vous me croyez si vous voulez, mais je vous assure que cette simple ligne fait le m\u00eame job que le programme pr\u00e9c\u00e9dent ! En mode console, on a:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; factorielle(7)\n5040<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">Python et factorielle: en utilisant une liste<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant que l\u2019on sait calculer une factorielle, on pourrait cr\u00e9er une liste des premi\u00e8res valeurs, mais de fa\u00e7on intelligente tant qu\u2019\u00e0 faire. Ainsi, on ne va pas \u00e9crire:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">F = []\nfor k in range(50):\n   F.append( factorielle(k) )<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ben non\u2026 Et je suis s\u00fbr que vous voyez pourquoi\u2026<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ben oui ! C\u2019est tout de m\u00eame ballot de faire toutes ces op\u00e9rations \u00e0 chaque fois que l\u2019on veut calculer un terme de la liste.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple, pour le&nbsp;<em>n<\/em>-i\u00e8me \u00e9l\u00e9ment de la liste, on demande de calculer&nbsp;<em>factorielle<\/em>(<em>n<\/em>&nbsp;\u2013 1), soit&nbsp;<em>n<\/em>&nbsp;\u2013 2 multiplications, plus une multiplication par&nbsp;<em>n<\/em>, alors qu\u2019il suffirait de multiplier l\u2019\u00e9l\u00e9ment pr\u00e9c\u00e9dent par (<em>n<\/em>&nbsp;\u2013 1).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Autrement dit, ce programme n\u00e9cessite $0+1+2+3+\\cdots+n=\\frac{n(n+1)}{2}$ multiplications (pour les connaisseurs, je ne parle pas de la complexit\u00e9 du programme, mais cette complexit\u00e9 est du m\u00eame ordre de grandeur), soit un ordre de grandeur de&nbsp;n\u00b2\/2&nbsp;multiplications.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Il est donc pr\u00e9f\u00e9rable d\u2019utiliser ce programme:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">F = [1]\nfor k in range(1,50):\n    F.append( F[k-1] * k )<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ce programme est bien plus rapide pour les grandes valeurs de&nbsp;<em>n<\/em>&nbsp;car il ne n\u00e9cessite que&nbsp;<em>n<\/em>&nbsp;produits (qui est bien moins grand que&nbsp;n\u00b2\/2).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si l\u2019on souhaite \u00e9crire une fonction, on pourra donc par exemple \u00e9crire ceci:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">global F\nF = [1]\n\ndef factorial(n):\n    if len(F) > n:\n        return F[n]\n    \n    for k in range(len(F),n+1):\n        F.append( F[k-1] * k )\n    \n    return F[n]<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Le fait de d\u00e9clarer la liste F en \u00ab&nbsp;global&nbsp;\u00bb permet de la garder en m\u00e9moire. Ainsi, lors du premier appel de la fonction&nbsp;<em>factorial(n)<\/em>, toutes les factorielles sont calcul\u00e9es jusqu\u2019\u00e0 n!, mais lors du deuxi\u00e8me appel, si l\u2019argument de la fonction est inf\u00e9rieur au premier&nbsp;<em>n<\/em>, il n\u2019y a pas de calculs suppl\u00e9mentaires. Et si l\u2019argument est sup\u00e9rieur \u00e0 la premi\u00e8re valeur de&nbsp;<em>n<\/em>, seules les factorielles qu\u2019il reste \u00e0 calculer le sont.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Python et factorielle: avec le module math<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Bien entendu, il existe le module&nbsp;<em>math<\/em>, d\u00e9di\u00e9 aux fonctions math\u00e9matiques, dans lequel est d\u00e9j\u00e0 impl\u00e9ment\u00e9e la fonction&nbsp;<em>factorial<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Gardons en t\u00eate que toutes les fonctions de ce module sont \u00e9crites en langage C, et donc leur ex\u00e9cution est tr\u00e8s rapide par rapport \u00e0 ce que l\u2019on peut impl\u00e9menter en Python.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Et n\u2019oubliez pas que si vous avez des probl\u00e8mes en maths,&nbsp;<a href=\"https:\/\/courspasquet.fr\/\" target=\"_blank\" rel=\"noreferrer noopener\">je peux vous aider par webcam<\/a>&nbsp;!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons voir sur cette page plusieurs mani\u00e8res de calculer une&nbsp;factorielle en Python. Qu&rsquo;est-ce qu&rsquo;une factorielle du point de vue math\u00e9matique ? Du point de vue math\u00e9matiques, la&nbsp;factorielle&nbsp;d\u2019un entier naturel&nbsp;n&nbsp;est le produit not\u00e9&nbsp;n! tel que : n!=1\u00d72\u00d73\u00d74\u00d7\u22ef\u00d7(n\u22121)\u00d7n.&nbsp; Par exemple, Python et factorielle: une approche na\u00efve \u00c0 partir de la d\u00e9finition [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-10756","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Plusieurs fa\u00e7ons de calculer une factorielle en Python - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Nous allons voir sur cette page plusieurs mani\u00e8res de calculer une\u00a0factorielle en Python. 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