{"id":8236,"date":"2023-04-10T16:33:17","date_gmt":"2023-04-10T14:33:17","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=8236"},"modified":"2023-04-10T16:33:18","modified_gmt":"2023-04-10T14:33:18","slug":"permutations-et-somme-des-resultats","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/","title":{"rendered":"Permutations et somme des r\u00e9sultats"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Int\u00e9ressons-nous \u00e0 un probl\u00e8me faisant intervenir les permutations dans lequel il nous est demand\u00e9 de d\u00e9terminer la somme des r\u00e9sultats.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#Permutations_et_somme_des_resultats_lenonce_du_probleme\" >Permutations et somme des r\u00e9sultats: l&#8217;\u00e9nonc\u00e9 du probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#Permutations_et_somme_des_resultats_premiere_approche_mathematique\" >Permutations et somme des r\u00e9sultats: premi\u00e8re approche math\u00e9matique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#Permutations_et_somme_des_resultats_approche_en_Python_du_probleme\" >Permutations et somme des r\u00e9sultats: approche en Python du probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#Approche_en_mathematique_du_probleme\" >Approche en math\u00e9matique du probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#Generalisation\" >G\u00e9n\u00e9ralisation<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Permutations_et_somme_des_resultats_lenonce_du_probleme\"><\/span>Permutations et somme des r\u00e9sultats: l&#8217;\u00e9nonc\u00e9 du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Le probl\u00e8me sur lequel nous allons nous pencher est expos\u00e9 dans cette vid\u00e9o. Je vais l&#8217;aborder diff\u00e9remment&#8230; et en fran\u00e7ais \ud83d\ude42<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Solving An Amazon Interview Puzzle - Sum of 5 digit numbers without repetition\" width=\"750\" height=\"422\" src=\"https:\/\/www.youtube.com\/embed\/wamTGA2R68s?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Soit S l&#8217;ensemble des nombres \u00e0 5 chiffres form\u00e9s en utilisant les chiffres 1, 2, 3, 4, 5 sans r\u00e9p\u00e9tition. Quelle est la somme de tous les \u00e9l\u00e9ments de S ?<\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Permutations_et_somme_des_resultats_premiere_approche_mathematique\"><\/span>Permutations et somme des r\u00e9sultats: premi\u00e8re approche math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il s&#8217;agit avant tout de savoir comment est constitu\u00e9 l&#8217;ensemble S. Pour les connaisseurs, il s&#8217;agit bien \u00e9videmment d&#8217;une permutation de l&#8217;ensemble {1,2,3,4,5}. Il y a donc 5! = 120 \u00e9l\u00e9ments dans S.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Pour les non initi\u00e9s, je vais tenter de vous expliquer la logique: commen\u00e7ons par un ensemble {1,2,3} et trouvons toutes les permutations de cet ensemble, c&#8217;est-\u00e0-dire tous les nombres \u00e0 3 chiffres compos\u00e9s des chiffres 1, 2 et 3 sans que ceux-ci se r\u00e9p\u00e8tent. On peut les \u00e9num\u00e9rer facilement:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-small-font-size\">ceux commen\u00e7ant par &#8220;1&#8221; : on a alors le choix entre les nombres &#8220;2&#8221; et &#8220;3&#8221; pour le deuxi\u00e8me chiffre, et il ne restera alors qu&#8217;un choix pour le dernier chiffre;<\/li>\n\n\n\n<li class=\"has-small-font-size\">ceux commen\u00e7ant par &#8220;2&#8221;: on a alors le choix entre les nombres &#8220;1&#8221; et &#8220;3&#8221; pour le deuxi\u00e8me chiffre, et il ne restera alors qu&#8217;un choix pour le dernier chiffre;<\/li>\n\n\n\n<li class=\"has-small-font-size\">ceux commen\u00e7ant par &#8220;3&#8221;: on a alors le choix entre les nombres &#8220;1&#8221; et &#8220;2&#8221; pour le deuxi\u00e8me chiffre, et il ne restera alors qu&#8217;un choix pour le dernier chiffre.<\/li>\n<\/ul>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut visualiser ces choix \u00e0 l&#8217;aide d&#8217;une repr\u00e9sentation graphique, que l&#8217;on appelle <em>arbre des possibilit\u00e9s<\/em>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"173\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-300x173.png\" alt=\"permutations somme r\u00e9sultats\" class=\"wp-image-8237\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-300x173.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-600x346.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-768x443.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image.png 876w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><figcaption class=\"wp-element-caption\">Arbre des possibilit\u00e9s des permutations de l&#8217;ensemble {1,2,3}<\/figcaption><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On voit qu&#8217;il y a 3 choix pour le premier chiffre, 2 pour le deuxi\u00e8me et 1 pour le dernier; le nombre de permutations est donc: \\(3\\times2\\times1=6\\). On note cela aussi: \\(6! = 3\\times2\\times1\\).<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Si on g\u00e9n\u00e9ralise ceci \u00e0 l&#8217;ensemble {1,2,3,4,5}, on trouve qu&#8217;il y a \\(5!=5\\times4\\times3\\times2\\times1=120\\) permutations. <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Les permutations font partie d&#8217;une branche des math\u00e9matiques que l&#8217;on appelle la <em><a href=\"https:\/\/fr.wikipedia.org\/wiki\/Combinatoire\" target=\"_blank\" rel=\"noreferrer noopener\">combinatoire<\/a>.<\/em> C&#8217;est la m\u00eame logique que les <em><a href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/08\/27\/anagrammes-et-python\/\" target=\"_blank\" rel=\"noreferrer noopener\">anagrammes<\/a><\/em>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Permutations_et_somme_des_resultats_approche_en_Python_du_probleme\"><\/span>Permutations et somme des r\u00e9sultats: approche en Python du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En Python, il existe le module <em>itertools<\/em> qui nous permet de construire les permutations de l&#8217;ensemble {1,2,3,4,5}. Il retourne les r\u00e9sultats sous la forme de 5-uples, qu&#8217;il nous faudra transformer en nombres afin de pouvoir en faire la somme. Pour cela, je vais utiliser la fonction <em>map<\/em>.<\/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=\"\">from itertools import permutations\n\nS = 0 # somme initiale: on part de z\u00e9ro\n\nfor i in permutations([1,2,3,4,5]):\n    S += int( ''.join( map(str,i) ) )\n    \nprint( S )<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Explications:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>d&#8217;abord, on parcourt toutes les permutations de l&#8217;ensemble {1, 2, 3, 4, 5} (mis pour l&#8217;occasion sous forme de liste) : c&#8217;est la boucle de la ligne 5;<\/li>\n\n\n\n<li>&#8220;map(str,i)&#8221; transforme l&#8217;item <em>i<\/em> (donc une permutation sous la forme d&#8217;un 5-uple) en une liste de caract\u00e8res;<\/li>\n\n\n\n<li>&#8220;join( map(str,i) )&#8221; transforme la liste de caract\u00e8res en cha\u00eene de caract\u00e8res;<\/li>\n\n\n\n<li>int( &#8221;.join( map(str,i) ) ) transforme la cha\u00eene de caract\u00e8res en nombre entier<\/li>\n\n\n\n<li>&#8220;S += int( &#8221;.join( map(str,i) ) )&#8221; est une autre syntaxe que &#8220;S = S + int( &#8221;.join( map(str,i) ) )&#8221; : on ajoute \u00e0 ce qui \u00e9tait dans la variable S l&#8217;entier obtenu<\/li>\n<\/ul>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">S repr\u00e9sente alors la somme de toutes les permutations. <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On trouve alors qu&#8217;l y en a 3 999 960.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Approche_en_mathematique_du_probleme\"><\/span>Approche en math\u00e9matique du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Quand un probl\u00e8me est complexe, j&#8217;ai souvent l&#8217;habitude de le simplifier au maximum pour en comprendre les rouages, et c&#8217;est ce que nous allons faire \u00e0 travers le premier exemple (permutation de l&#8217;ensemble {1,2,3}).<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Les nombres obtenus par permutations de {1,2,3} constituent un ensemble S et sont de la forme:$$\\overline{x_2x_1x_0}=100x_2+10x_1+x_0.$$ Par exemple,$$123=100\\times1+10\\times2+3.$$On cherche donc \u00e0 ajouter tous ces nombres:$$\\sum_{S}100x_2+10x_1+x_0=100\\sum_{S}x_2+10\\sum_{S}x_1+\\sum_{S}x_0.$$Or, $$\\sum_{S}x_2=\\sum_{S}x_1=\\sum_{S}x_0$$car tous les chiffres apparaissent un m\u00eame nombre de fois pour chacune des positions dans le nombre. La somme vaut donc:$$(100+10+1)\\sum_{S}x_0=111\\sum_{S}x_0.$$De plus,$$\\sum_{S}x_0=2\\times(1+2+3)=12$$car chaque chiffre appara\u00eet ici deux fois.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">La somme cherch\u00e9e sur notre exemple est donc:$$111\\times12=1332.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Revenons maintenant \u00e0 la somme des permutations de {1,2,3,4,5}. Avec les m\u00eames notations que pr\u00e9c\u00e9demment, on obtient que la somme est:$$(10^5+10^4+10^3+10^2+10+1)\\sum_{S}x_0=11111\\sum_{S}x_0.$$Il nous faut maintenant trouver combien chaque chiffre appara\u00eet \u00e0 une position donn\u00e9e. Il suffit de compter combien de &#8220;1&#8221; apparaissent en premi\u00e8re position par exemple: combien de permutations commencent par &#8220;1&#8221; sachant qu&#8217;il reste 4 chiffres? Il y en a 4!=24.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Maintenant, on calcule la somme:$$5+4+3+2+1=15.$$La somme de toutes les permutations est alors:$$11111\\times15\\times24=3999960.$$Le compte est bon!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generalisation\"><\/span>G\u00e9n\u00e9ralisation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Quelle est la somme de toutes les permutations de l&#8217;ensemble {1,2,3,&#8230;,<em>n<\/em>} ?<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">D&#8217;apr\u00e8s ce qui a \u00e9t\u00e9 vu pr\u00e9c\u00e9demment, on peut dire que cette somme est:$$\\left(\\sum_{k=0}^n10^k\\right) \\times \\left(\\sum_{k=1}^n k\\right) \\times (n-1)!.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Int\u00e9ressons-nous \u00e0 un probl\u00e8me faisant intervenir les permutations dans lequel il nous est demand\u00e9 de d\u00e9terminer la somme des r\u00e9sultats.<\/p>\n","protected":false},"author":1,"featured_media":8245,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[204],"class_list":["post-8236","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-arbre"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Permutations et somme des r\u00e9sultats - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l&#039;ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Permutations et somme des r\u00e9sultats - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l&#039;ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2023-04-10T14:33:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-04-10T14:33:18+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"740\" \/>\n\t<meta property=\"og:image:height\" content=\"198\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Permutations et somme des r\u00e9sultats\",\"datePublished\":\"2023-04-10T14:33:17+00:00\",\"dateModified\":\"2023-04-10T14:33:18+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/\"},\"wordCount\":873,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/04\\\/head-articles-maths-permutations.jpg\",\"keywords\":[\"arbre\"],\"articleSection\":[\"Math\u00e9matiques\",\"Python\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/\",\"name\":\"Permutations et somme des r\u00e9sultats - Mathweb.fr\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/04\\\/head-articles-maths-permutations.jpg\",\"datePublished\":\"2023-04-10T14:33:17+00:00\",\"dateModified\":\"2023-04-10T14:33:18+00:00\",\"description\":\"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l'ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/04\\\/head-articles-maths-permutations.jpg\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2023\\\/04\\\/head-articles-maths-permutations.jpg\",\"width\":740,\"height\":198,\"caption\":\"permutations somme r\u00e9sultat\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2023\\\/04\\\/10\\\/permutations-et-somme-des-resultats\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Permutations et somme des r\u00e9sultats\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\",\"name\":\"Mathweb.fr\",\"description\":\"Math\u00e9matiques, LaTeX et Python\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\",\"name\":\"St\u00e9phane Pasquet\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"width\":74,\"height\":77,\"caption\":\"St\u00e9phane Pasquet\"},\"logo\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Permutations et somme des r\u00e9sultats - Mathweb.fr","description":"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l'ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/","og_locale":"fr_FR","og_type":"article","og_title":"Permutations et somme des r\u00e9sultats - Mathweb.fr","og_description":"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l'ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,","og_url":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/","og_site_name":"Mathweb.fr","article_published_time":"2023-04-10T14:33:17+00:00","article_modified_time":"2023-04-10T14:33:18+00:00","og_image":[{"width":740,"height":198,"url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg","type":"image\/jpeg"}],"author":"St\u00e9phane Pasquet","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"St\u00e9phane Pasquet","Dur\u00e9e de lecture estim\u00e9e":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#article","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/"},"author":{"name":"St\u00e9phane Pasquet","@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"headline":"Permutations et somme des r\u00e9sultats","datePublished":"2023-04-10T14:33:17+00:00","dateModified":"2023-04-10T14:33:18+00:00","mainEntityOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/"},"wordCount":873,"commentCount":0,"publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg","keywords":["arbre"],"articleSection":["Math\u00e9matiques","Python"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/","url":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/","name":"Permutations et somme des r\u00e9sultats - Mathweb.fr","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#primaryimage"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg","datePublished":"2023-04-10T14:33:17+00:00","dateModified":"2023-04-10T14:33:18+00:00","description":"Quelle est la somme de toutes les r\u00e9sultats obtenus par permutations de l'ensemble {1,2,3,4,5} ? Approche math\u00e9matique et Python,","breadcrumb":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#primaryimage","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/head-articles-maths-permutations.jpg","width":740,"height":198,"caption":"permutations somme r\u00e9sultat"},{"@type":"BreadcrumbList","@id":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/10\/permutations-et-somme-des-resultats\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.mathweb.fr\/euclide\/"},{"@type":"ListItem","position":2,"name":"Permutations et somme des r\u00e9sultats"}]},{"@type":"WebSite","@id":"https:\/\/www.mathweb.fr\/euclide\/#website","url":"https:\/\/www.mathweb.fr\/euclide\/","name":"Mathweb.fr","description":"Math\u00e9matiques, LaTeX et Python","publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.mathweb.fr\/euclide\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":["Person","Organization"],"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69","name":"St\u00e9phane Pasquet","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","width":74,"height":77,"caption":"St\u00e9phane Pasquet"},"logo":{"@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp"}}]}},"_links":{"self":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/8236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/comments?post=8236"}],"version-history":[{"count":0,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/8236\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media\/8245"}],"wp:attachment":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media?parent=8236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/categories?post=8236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/tags?post=8236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}