{"id":233339,"date":"2022-03-30T17:07:31","date_gmt":"2022-03-30T15:07:31","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=233339"},"modified":"2025-09-29T11:54:18","modified_gmt":"2025-09-29T09:54:18","slug":"denombrement-exercice-corrige-mpsi","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/denombrement-exercice-corrige-mpsi\/","title":{"rendered":"D\u00e9nombrement : exercice corrig\u00e9 – MPSI"},"content":{"rendered":"\n

Tu connais ton cours sur le d\u00e9nombrement<\/strong> ? La prochaine \u00e9tape pour ma\u00eetriser cette notion, c’est de t’entra\u00eener avec nos exercices corrig\u00e9s<\/strong> ! Apr\u00e8s avoir fait cela, le d\u00e9nombrement<\/strong> n’aura plus aucun secret pour toi !<\/p>\n\n\n\n

Et si tu veux aller encore plus loin, transforme ton appr\u00e9hension du d\u00e9nombrement en confiance absolue gr\u00e2ce \u00e0 des cours particuliers de maths en visio<\/strong><\/a> structur\u00e9s et interactifs. \ud83d\ude80<\/p>\n\n\n\n

Exercice d’application sur le d\u00e9nombrement<\/h2>\n\n\n\n

\u23f0 Dur\u00e9e : 15 min<\/p>\n\n\n\n

\ud83d\udcaa Difficult\u00e9 : 1\/3 <\/p>\n\n\n\n\nSoit \"E\" un ensemble \u00e0 \"n\" \u00e9l\u00e9ments. D\u00e9nombrer le nombre de couples \"(A,B)\in\mathcal{P}(E)^2\" tels que \"A\cap B=\O\".\n\n\n\n

Corrig\u00e9 de l’exercice d’application<\/h2>\n\n\n\n
<\/div>\n\n\n\n \nSoit \"G\"={\"(A,B)\in\mathcal{P}(E)^2, A\cap B=\O\"}.\n\n\n\n
<\/div>\n\n\n\n \nPour \"k\in\mathsbb[\![0;n]\!]}\", on pose \"G_k\" = {\"(A,B)\in\mathcal{P}(E)^2, A\cap B=\O\" et \"card(A)=k\"}. \n
  • Soit \"(k,k')\in\mathsbb[\![0;n]\!]^2}\" avec \"k\ne k'\". On suppose qu’il existe \"(A,B)\in\mathsbb{G_k}\cap G_{k'}\". On aurait \"card(A) = k = k'\", ce qui est impossible. Ainsi, si \"k\ne k'\", \"{G_k}\cap G_{k'} = \O\".<\/li>\n
  • Montrons que \"\bigcup_{k=0}^n G_k = G\". L’inclusion \"\bigcup_{k=0}^n G_k \subset G\" est \u00e9vidente.<\/li>\n\n\n\n
    <\/div>\n\n\n\n \nSoit \"(A,B)\in\mathsbb{G_k}\", on commence par choisir une partie \"A\" de \"E\" \u00e0 \"k\" \u00e9l\u00e9ments, ce que l’on peut faire de \"\binom{n}{k}\" fa\u00e7ons, puis on choisit \"B\in\mathcal{P}(E\backslash A)\", ce que l’on peut faire de \"2^{n-k}\" fa\u00e7ons car \"card(E\backslash A)=n-k\".\n\n\n\n
    <\/div>\n\n\n\n\nPar la formule du bin\u00f4me, on a :\n

      <\/span>   <\/span>\"\[<\/p>\n\n\n\n

    \n
    \n
    \"livre<\/figure>\n<\/div>\n\n\n\n
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    Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n

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    \n Tu as aim\u00e9 cet article ?<\/span>\n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

    Tu connais ton cours sur le d\u00e9nombrement ? La prochaine \u00e9tape pour ma\u00eetriser cette notion, c’est de t’entra\u00eener (…)<\/p>\n","protected":false},"author":158,"featured_media":244614,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-233339","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/233339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=233339"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/233339\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244614"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=233339"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=233339"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=233339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}