{"id":7213,"date":"2011-11-27T21:18:46","date_gmt":"2011-11-27T21:18:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7213"},"modified":"2022-01-25T22:54:14","modified_gmt":"2022-01-25T22:54:14","slug":"um-grupo-de-doze-rapazes-e-oito-raparigas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7213","title":{"rendered":"Um grupo de doze rapazes e oito raparigas"},"content":{"rendered":"<p><ul id='GTTabs_ul_7213' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7213' class='GTTabs_curr'><a  id=\"7213_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7213' ><a  id=\"7213_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7213'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7214\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7214\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb.jpg\" data-orig-size=\"500,235\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1243022964&quot;,&quot;copyright&quot;:&quot;Kurhan - Fotolia&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;young smiling  people. Isolated over white background&quot;}\" data-image-title=\"Jovens\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb.jpg\" class=\"alignright size-full wp-image-7214\" title=\"Jovens\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb.jpg\" alt=\"\" width=\"300\" height=\"141\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb.jpg 500w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb-300x141.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb-150x70.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/5jovensb-400x188.jpg 400w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um grupo de doze rapazes e oito raparigas pretende organizar um clube.<\/p>\n<ol>\n<li>De quantos modos diferentes se pode obter uma dire\u00e7\u00e3o de cinco elementos com fun\u00e7\u00f5es indiferenciadas, sabendo que:<br \/>\na) s\u00e3o todos eleg\u00edveis;<br \/>\nb) \u00e9 formada s\u00f3 por rapazes;<br \/>\nc) \u00e9 formada s\u00f3 por raparigas;<br \/>\nd) \u00e9 formada por 3 rapazes e duas raparigas?<\/li>\n<li>Sabendo que a escolha dos elementos para a dire\u00e7\u00e3o \u00e9 feita por sorteio e que todos s\u00e3o eleg\u00edveis, qual a probabilidade:<br \/>\na) dos cinco elementos serem todos rapazes;<br \/>\nb) da dire\u00e7\u00e3o ter, no m\u00e1ximo, dois rapazes?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7213' onClick='GTTabs_show(1,7213)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7213'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>O n\u00famero de modos diferentes que se pode obter uma dire\u00e7\u00e3o de cinco elementos com fun\u00e7\u00f5es indiferenciadas, sabendo que:\n<p>a) s\u00e3o todos eleg\u00edveis, \u00e9 ${}^{20}{{C}_{5}}=15504$;<\/p>\n<p>b) \u00e9 formada s\u00f3 por rapazes, \u00e9 ${}^{12}{{C}_{5}}=792$;<\/p>\n<p>c) \u00e9 formada s\u00f3 por raparigas, \u00e9 ${}^{8}{{C}_{5}}=56$;<\/p>\n<p>d) \u00e9 formada por 3 rapazes e duas raparigas, \u00e9 ${}^{12}{{C}_{3}}\\times {}^{8}{{C}_{2}}=220\\times 28=6160$.<\/p>\n<\/li>\n<li>Sabendo que a escolha dos elementos para a dire\u00e7\u00e3o \u00e9 feita por sorteio e que todos s\u00e3o eleg\u00edveis, a probabilidade:\n<p>a) dos cinco elementos serem todos rapazes, \u00e9 $$p=\\frac{{}^{12}{{C}_{5}}}{{}^{20}{{C}_{5}}}=\\frac{792}{15504}=\\frac{33}{646}\\approx 0,051$$<\/p>\n<p>b) da dire\u00e7\u00e3o ter, no m\u00e1ximo, dois rapazes, \u00e9 $$p=\\frac{{}^{12}{{C}_{0}}\\times {}^{8}{{C}_{5}}+{}^{12}{{C}_{1}}\\times {}^{8}{{C}_{4}}+{}^{12}{{C}_{2}}\\times {}^{8}{{C}_{3}}}{{}^{20}{{C}_{5}}}=\\frac{1\\times 56+12\\times 70+66\\times 56}{15504}=\\frac{4592}{15504}=\\frac{287}{969}\\approx 0,296$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7213' onClick='GTTabs_show(0,7213)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um grupo de doze rapazes e oito raparigas pretende organizar um clube. De quantos modos diferentes se pode obter uma dire\u00e7\u00e3o de cinco elementos com fun\u00e7\u00f5es indiferenciadas, sabendo que: a) s\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21041,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,255],"series":[],"class_list":["post-7213","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-analise-combinatoria"],"views":2149,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag175-52_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7213"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7213\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21041"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7213"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7213"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7213"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}