{"id":2586,"date":"2022-04-30T11:47:07","date_gmt":"2022-04-30T09:47:07","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/permutace-s-opakovanim-zakladni-vzorec-vzorovy-priklad\/"},"modified":"2022-04-30T12:47:59","modified_gmt":"2022-04-30T10:47:59","slug":"permutace-s-opakovanim-zakladni-vzorec-vzorovy-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/permutace-s-opakovanim-zakladni-vzorec-vzorovy-priklad\/","title":{"rendered":"Permutace s opakov\u00e1n\u00edm &#8211; z\u00e1kladn\u00ed vzorec + vzorov\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p>Zat\u00edmco v p\u0159\u00edpad\u011b Permutac\u00ed bez opakov\u00e1n\u00ed jsme uspo\u0159\u00e1d\u00e1vali permutace z dan\u00fdch prvk\u016f, p\u0159i\u010dem\u017e ka\u017ed\u00fd se v dan\u00fd permutaci vyskytoval pouze jednou (kolika zp\u016fsoby se\u0159ad\u00edme na poli\u010dku 4 knihy, kolika zp\u016fsoby se\u0159ad\u00edme do \u0159ady 5 d\u011bt\u00ed atd.), pokud po\u010d\u00edt\u00e1me PERMUTACE S OPAKOV\u00c1N\u00cdM, <strong>n\u011bkter\u00e9 prvky se mohou opakovat<\/strong>. Ale pozor! Na jejich po\u0159ad\u00ed n\u00e1m st\u00e1le z\u00e1le\u017e\u00ed.<\/p>\n<ul>\n<li>Vzorec pro v\u00fdpo\u010det PERMUTAC\u00cd S OPAKOV\u00c1N\u00cdM vyu\u017eijeme v p\u0159\u00edpad\u011b, kdy chceme vypo\u010d\u00edtat kolik zp\u016fsoby m\u016f\u017eeme za lokomotivu uspo\u0159\u00e1dat 3 vag\u00f3ny, kdy\u017e dva z nich jsou osobn\u00ed a jeden n\u00e1kladn\u00ed (ty osobn\u00ed vag\u00f3ny mezi sebou nerozli\u0161ujeme).<\/li>\n<li>PERMUTACE S OPAKOV\u00c1N\u00cdM vyu\u017eijeme tak\u00e9 v p\u0159\u00edpad\u011b, kdy chceme zjistit kolika zp\u016fsoby se daj\u00ed p\u0159eskupit p\u00edsmena nap\u0159\u00edklad ve slov\u011b ABRAKADABRA.<\/li>\n<li>P\u0159i po\u010d\u00edt\u00e1n\u00ed PERMUTAC\u00cd S OPAKOV\u00c1N\u00cdM <strong>mus\u00edme v\u017edy zohlednit prvky, kter\u00e9 mezi sebou nerozli\u0161ujeme.\u00a0<\/strong>Poj\u010fme si na konkr\u00e9tn\u00edm p\u0159\u00edpadu uk\u00e1zat, co t\u00edm mysl\u00edme. V p\u0159\u00edpad\u011b p\u0159ehazov\u00e1n\u00ed p\u00edsmen ve slov\u011b ABRAKADABRA mus\u00edme vz\u00edt v potaz to, \u017ee:<\/li>\n<\/ul>\n<ol>\n<li>p\u00edsmeno <strong>A<\/strong> se v tomto slov\u011b vyskytuje <strong>5x<\/strong><\/li>\n<li>p\u00edsmeno<strong> B<\/strong> se v tomto slov\u011b vyskytuje <strong>2x<\/strong><\/li>\n<li>p\u00edsmeno<strong> R <\/strong>se v tomto slov\u011b vyskytuje <strong>2x<\/strong><\/li>\n<li>p\u00edsmeno<strong> K<\/strong> se v tomto slov\u011b vyskytuje <strong>1x<\/strong><\/li>\n<li>p\u00edsmeno<strong> D<\/strong>\u00a0se v tomto slov\u011b vyskytuje <strong>1x<\/strong><\/li>\n<\/ol>\n<ul>\n<li>Slovo ABRAKADABRA je tvo\u0159eno z 5 r\u016fzn\u00fdch p\u00edsmen.<\/li>\n<li>Z\u00e1kladn\u00ed vzorec pro v\u00fdpo\u010det PERMUTAC\u00cd S OPAKOV\u00c1N\u00cdM vypad\u00e1 n\u00e1sledovn\u011b:<\/li>\n<\/ul>\n<p>$$P\u00b4(k1,k2,&#8230;kn)=\\frac{(k1+k2+&#8230;.+kn)!}{k1!k2!&#8230;.kn!}$$<\/p>\n<ul>\n<li><em><strong>k<\/strong><\/em> zna\u010d\u00ed kolika mo\u017enostmi m\u016f\u017ee b\u00fdt prvek p\u0159e\u0159azen<\/li>\n<li>Vzorec a to jak jej spr\u00e1vn\u011b pou\u017e\u00edvat si vysv\u011btl\u00edme na na\u0161em p\u0159\u00edkladu ABRAKADABRA.<\/li>\n<li>5xA = k1 = 5<\/li>\n<li>2XB = k2 = 2<\/li>\n<li>2xR = k3 = 2<\/li>\n<li>1xK = k4 = 1<\/li>\n<li>1xD = k5 = 1<\/li>\n<\/ul>\n<p style=\"text-align: center\">Nyn\u00ed zb\u00fdv\u00e1 dosadit do vzorce pro v\u00fdpo\u010det Permutac\u00ed s opakov\u00e1n\u00edm<\/p>\n<p>$$P\u00b4(k1,k2,&#8230;kn)=\\frac{(k1+k2+&#8230;.+kn)!}{k1!k2!&#8230;.kn!}$$<\/p>\n<p>$$P\u00b4(5,2,2,1,1)=\\frac{(5+2+2+1+1)!}{5!2!2!1!1!}=\\frac{11!}{5!2!2!1!1!}=83160$$<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2586","lesson","type-lesson","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Permutace s opakov\u00e1n\u00edm - z\u00e1kladn\u00ed vzorec + vzorov\u00fd p\u0159\u00edklad | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/permutace-s-opakovanim-zakladni-vzorec-vzorovy-priklad\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Permutace s opakov\u00e1n\u00edm - z\u00e1kladn\u00ed vzorec + vzorov\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zat\u00edmco v p\u0159\u00edpad\u011b Permutac\u00ed bez opakov\u00e1n\u00ed jsme uspo\u0159\u00e1d\u00e1vali permutace z dan\u00fdch prvk\u016f, p\u0159i\u010dem\u017e ka\u017ed\u00fd se v dan\u00fd permutaci vyskytoval pouze jednou (kolika zp\u016fsoby se\u0159ad\u00edme na poli\u010dku 4 knihy, kolika zp\u016fsoby se\u0159ad\u00edme do \u0159ady 5 d\u011bt\u00ed atd.), pokud po\u010d\u00edt\u00e1me PERMUTACE S OPAKOV\u00c1N\u00cdM, n\u011bkter\u00e9 prvky se mohou opakovat. 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