{"id":7232,"date":"2011-12-02T21:35:26","date_gmt":"2011-12-02T21:35:26","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7232"},"modified":"2022-01-25T23:32:01","modified_gmt":"2022-01-25T23:32:01","slug":"coca-cola-e-ice-tea","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7232","title":{"rendered":"Latas de Coca-Cola e Ice Tea"},"content":{"rendered":"<p><ul id='GTTabs_ul_7232' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7232' class='GTTabs_curr'><a  id=\"7232_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7232' ><a  id=\"7232_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_7232'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7233\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7233\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton.jpg\" data-orig-size=\"302,275\" 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;0&quot;,&quot;copyright&quot;:&quot;&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;&quot;}\" data-image-title=\"Latas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton.jpg\" class=\"alignright size-full wp-image-7233\" title=\"Latas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton.jpg\" alt=\"\" width=\"181\" height=\"165\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton.jpg 302w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton-300x273.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/cola-lipton-150x136.jpg 150w\" sizes=\"auto, (max-width: 181px) 100vw, 181px\" \/><\/a>A professora de Matem\u00e1tica do 12.\u00ba X prop\u00f4s o seguinte problema \u00e0 turma:<\/p>\n<p>&#8220;<em>Uma grade tem doze compartimentos para colocar latas de refrigerantes. De quantas formas diferentes podemos arrumar sete latas na grade, sabendo que quatro delas s\u00e3o de Coca-Cola (e, portanto, indistingu\u00edveis) e as restantes s\u00e3o de Ice Tea (uma de lim\u00e3o, uma de p\u00eassego e outra de manga)<\/em>.&#8221;<\/p>\n<p>A Maria e o Pedro foram os primeiros a responder com seguran\u00e7a. Os resultados que apresentaram est\u00e3o ambos certos e foram os seguintes:<\/p>\n<ul>\n<li>Maria: ${}^{12}{{C}_{7}}\\times {}^{7}{{A}_{3}}$<\/li>\n<li>Pedro: ${}^{12}{{C}_{4}}\\times {}^{8}{{A}_{3}}$<\/li>\n<\/ul>\n<p>Numa composi\u00e7\u00e3o de dez a quinze linhas, explique o racioc\u00ednio de cada um dos alunos.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7232' onClick='GTTabs_show(1,7232)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7232'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ul>\n<li>Coca-Cola: 4 latas indistingu\u00edveis;<\/li>\n<li>Ice Tea: 3 latas distingu\u00edveis (lim\u00e3o, p\u00eassego e manga).<\/li>\n<\/ul>\n<p><strong>Maria<\/strong>:<\/p>\n<p>As sete latas de refrigerante v\u00e3o ocupar sete dos doze compartimentos da grade.<\/p>\n<p>Estes compartimentos podem ser selecionados de ${}^{12}{{C}_{7}}$ modos diferentes.<\/p>\n<p>Apresenta-se, seguidamente, uma dessas poss\u00edveis sele\u00e7\u00f5es:<\/p>\n<table class=\" aligncenter\" style=\"width: 30%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>1<\/sub><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>2<\/sub><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>3<\/sub><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>4<\/sub><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>5<\/sub><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>6<\/sub><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">s<sub>7<\/sub><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Para esta sele\u00e7\u00e3o dos compartimentos da grade, comecemos por pensar na coloca\u00e7\u00e3o das latas de Ice Tea por ordem de sabor, por exemplo, lim\u00e3o, p\u00eassego e manga. Dois resultados poss\u00edveis ser\u00e3o, por exemplo, $\\left( {{s}_{4}},{{s}_{7}},{{s}_{1}} \\right)$ e $\\left( {{s}_{7}},{{s}_{4}},{{s}_{1}} \\right)$. Para cada uma destas situa\u00e7\u00f5es, as quatro latas de Coca-Cola, porque s\u00e3o indistingu\u00edveis, ocupar\u00e3o os restantes quatro compartimentos livres, previamente selecionados.<\/p>\n<p>Portanto, para a sele\u00e7\u00e3o dos 7 compartimentos exemplificada, as 3 latas de Ice Tea (e as 4 de Coca-Cola, nos restantes 4 compartimentos que restarem livres) podem ser dispostas de $7\\times 6\\times 5={}^{7}{{A}_{3}}$ maneiras diferentes.<\/p>\n<p>Logo, considerando os ${}^{12}{{C}_{7}}$ modos diferentes de selecionar os sete compartimentos que as sete latas v\u00e3o ocupar, resulta que as latas podem ser arrumadas de ${}^{12}{{C}_{7}}\\times {}^{7}{{A}_{3}}$ maneiras diferentes.<\/p>\n<p><strong>Pedro<\/strong>:<\/p>\n<p>Comecemos por colocar as quatro latas de Coca-Cola.<\/p>\n<p>Como estas latas s\u00e3o indistingu\u00edveis, selecionados quatro dos doze compartimentos, existe apenas uma maneira de as arrumar na grade.<\/p>\n<p>Sendo assim, existem apenas ${}^{12}{{C}_{4}}$ maneiras diferentes de arrumar as latas de Coca-Cola. Consideremos uma dessas maneiras, por exemplo:<\/p>\n<table class=\" aligncenter\" style=\"width: 30%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">c<\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">\u00a0c<\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">c<\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\">\u00a0c<\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<td style=\"text-align: center; width: 20px; border: #a52a2a 1px solid;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Pensemos agora na coloca\u00e7\u00e3o das latas de Ice Tea por ordem de sabor, por exemplo, lim\u00e3o, p\u00eassego e manga. Temos vagos 8 dos doze compartimentos da grade, pelo que essas tr\u00eas latas podem ser dispostas de $8\\times 7\\times 6={}^{8}{{A}_{3}}$ maneiras diferentes.<\/p>\n<p>Logo, considerando os ${}^{12}{{C}_{4}}$ modos diferentes colocar as quatro latas de Coca-Cola, resulta que as latas podem ser arrumadas de\u00a0${}^{12}{{C}_{4}}\\times {}^{8}{{A}_{3}}$ maneiras diferentes.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7232' onClick='GTTabs_show(0,7232)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A professora de Matem\u00e1tica do 12.\u00ba X prop\u00f4s o seguinte problema \u00e0 turma: &#8220;Uma grade tem doze compartimentos para colocar latas de refrigerantes. De quantas formas diferentes podemos arrumar sete latas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21053,"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-7232","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":3197,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag178-62_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7232","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=7232"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7232\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21053"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7232"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7232"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7232"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7232"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}