{"id":2493,"date":"2022-04-25T16:44:35","date_gmt":"2022-04-25T16:44:35","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/kombinace-bez-opakovani-zakladni-vzorec-vzorovy-priklad\/"},"modified":"2022-04-26T20:32:34","modified_gmt":"2022-04-26T18:32:34","slug":"kombinace-bez-opakovani-zakladni-vzorec-vzorovy-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/kombinace-bez-opakovani-zakladni-vzorec-vzorovy-priklad\/","title":{"rendered":"Kombinace bez opakov\u00e1n\u00ed &#8211; z\u00e1kladn\u00ed vzorec + vzorov\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p>KOMBINACE BEZ OPAKOV\u00c1N\u00cd vyu\u017eijeme v p\u0159\u00edpad\u011b, kdy vyb\u00edr\u00e1me n\u011bjak\u00e9 prvky, ale nez\u00e1le\u017e\u00ed n\u00e1m na jejich po\u0159ad\u00ed.<\/p>\n<ul>\n<li>Kombinace bez opakov\u00e1n\u00ed jsou podobn\u00e9 variac\u00edm bez opakov\u00e1n\u00ed, ale <strong>z\u00e1sadn\u00ed rozd\u00edl<\/strong> je v tom, \u017ee u variac\u00ed bez opakov\u00e1n\u00ed n\u00e1m z\u00e1le\u017e\u00ed na po\u0159ad\u00ed prvk\u016f, <strong>u kombinac\u00ed bez opakov\u00e1n\u00ed n\u00e1m na po\u0159ad\u00ed prvk\u016f nez\u00e1le\u017e\u00ed!<\/strong><\/li>\n<li>Vzorec pro v\u00fdpo\u010det kombinac\u00ed bez opakov\u00e1n\u00ed vyu\u017eijeme nap\u0159\u00edklad p\u0159i v\u00fdb\u011br\u016f lid\u00ed do t\u00fdm\u016f (nez\u00e1le\u017e\u00ed v jak\u00e9m po\u0159ad\u00ed vybereme \u010dleny), p\u0159i zp\u016fsobu vyb\u00edr\u00e1n\u00ed \u010d\u00edsel do sportky (op\u011bt nez\u00e1le\u017e\u00ed na po\u0159ad\u00ed \u010d\u00edsel), vyb\u00edr\u00e1n\u00ed neuspo\u0159\u00e1dan\u00fdch skupin a podobn\u011b.<\/li>\n<li>Z\u00e1kladn\u00ed vzorec pro v\u00fdpo\u010det kombinac\u00ed bez opakov\u00e1n\u00ed vypad\u00e1:<\/li>\n<\/ul>\n<p>$$K(k,n)=\\frac{n!}{k!(n-k)!}$$<\/p>\n<p>Tomuto vzorci se tak\u00e9 \u0159\u00edk\u00e1 kombina\u010dn\u00ed \u010d\u00edslo a d\u00e1 se zapsat takto:<\/p>\n<p>$$\\binom{n}{k}$$<\/p>\n<p>\u010dte se jako <em><strong>n<\/strong><\/em> nad<em><strong> k\u00a0<\/strong><\/em><\/p>\n<ul>\n<li>Vzorec pro v\u00fdpo\u010det kombinac\u00ed bez opakov\u00e1n\u00ed je v podstat\u011b vzore\u010dek pro v\u00fdpo\u010det variac\u00ed bez opakov\u00e1n\u00ed vyn\u00e1soben\u00fd zlomkem jedna lomeno <strong><em>k!<\/em><\/strong><\/li>\n<\/ul>\n<p>$$K(k,n)=\\frac{1}{k!}\\cdot{V(k,n)}=\\frac{1}{k!}\\cdot{\\frac{n!}{(n-k)!}}=\\frac{n!}{k!(n-k)!}$$<\/p>\n<p>Vzorov\u00fd p\u0159\u00edklad:<\/p>\n<p>Na kru\u017enici je rozm\u00edst\u011bno 9 bod\u016f. Kolik existuje r\u016fzn\u00fdch troj\u00faheln\u00edk\u016f, jejich\u017e vrcholy jsou tyto body?<\/p>\n<p>V tomto p\u0159\u00edpad\u011b vyb\u00edr\u00e1me 3 mo\u017enosti z 9 a nez\u00e1le\u017e\u00ed n\u00e1m na jejich po\u0159ad\u00ed. Ur\u010d\u00edme si <strong><em>n<\/em><\/strong> a <strong><em>k<\/em><\/strong>.<\/p>\n<p><em><strong>n=9<\/strong><\/em>, <strong><em>k= 3<\/em><\/strong> a dosad\u00edme do vzorce pro v\u00fdpo\u010det kombinac\u00ed bez opakov\u00e1n\u00ed:<\/p>\n<p>$$K(k,n)=\\binom{n}{k}$$<\/p>\n<p>$$K(3,9)=\\binom{9}{3}=\\frac{9!}{3!6!}=\\frac{9\\cdot{8}\\cdot{7}\\cdot{6!}}{6!\\cdot{6}}=\\frac{9\\cdot{8}\\cdot{7}}{6}=84$$<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2493","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>Kombinace bez opakov\u00e1n\u00ed - 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\/kombinace-bez-opakovani-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=\"Kombinace bez opakov\u00e1n\u00ed - z\u00e1kladn\u00ed vzorec + vzorov\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"KOMBINACE BEZ OPAKOV\u00c1N\u00cd vyu\u017eijeme v p\u0159\u00edpad\u011b, kdy vyb\u00edr\u00e1me n\u011bjak\u00e9 prvky, ale nez\u00e1le\u017e\u00ed n\u00e1m na jejich po\u0159ad\u00ed. 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