{"id":7226,"date":"2011-12-02T18:25:00","date_gmt":"2011-12-02T18:25:00","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7226"},"modified":"2022-01-25T23:17:42","modified_gmt":"2022-01-25T23:17:42","slug":"numerar-as-faces-de-um-poliedro","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7226","title":{"rendered":"Numerar as faces de um poliedro"},"content":{"rendered":"<p><ul id='GTTabs_ul_7226' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7226' class='GTTabs_curr'><a  id=\"7226_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7226' ><a  id=\"7226_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_7226'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7227\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7227\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\" data-orig-size=\"332,224\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Poliedro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\" class=\"alignright size-full wp-image-7227\" title=\"Poliedro\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\" alt=\"\" width=\"199\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg 332w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59-300x202.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59-150x101.jpg 150w\" sizes=\"auto, (max-width: 199px) 100vw, 199px\" \/><\/a>Na figura est\u00e1 representado um poliedro com nove faces, que pode ser decomposto num cubo e numa pir\u00e2mide quadrangular regular.<\/p>\n<p>Pretende-se numerar as nove faces do poliedro com os n\u00fameros de 1 a 9 (um n\u00famero diferente em cada face).<\/p>\n<p>Como se v\u00ea na figura, tr\u00eas das faces do poliedro j\u00e1 est\u00e3o numeradas, com os n\u00fameros 1, 2 e 3.<\/p>\n<ol>\n<li>De quantas maneiras diferentes podemos numerar as outras seis faces, com os restantes seis n\u00fameros?<\/li>\n<li>Com os restantes seis n\u00fameros, de quantas maneiras podemos numerar as outras seis faces, de forma que, nas cinco faces do cubo, fiquem n\u00fameros cuja soma seja \u00edmpar?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7226' onClick='GTTabs_show(1,7226)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7226'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" title=\"Poliedro\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-59.jpg\" alt=\"\" width=\"199\" height=\"134\" \/><\/a>H\u00e1 seis possibilidades de escolher um dos n\u00fameros dispon\u00edveis para colocar na primeira face, cinco possibilidades para escolher outro n\u00famero para colocar na segunda face, e assim sucessivamente.\n<p>Logo, podemos numerar as outras seis faces, com os restantes seis n\u00fameros, de $N={}^{6}{{A}_{6}}={{P}_{6}}=6!=6\\times 5\\times 4\\times 3\\times 2\\times 1=720$ maneiras diferentes.<\/p>\n<\/li>\n<li>Os seis n\u00fameros dispon\u00edveis s\u00e3o: 4, 5, 6, 7, 8 e 9.\n<p>Faltam numerar 4 das cinco faces acess\u00edveis do cubo, pois uma delas j\u00e1 est\u00e1 numerada com o n\u00famero 2.<\/p>\n<p>Para que a soma dos n\u00fameros dessas faces d\u00ea \u00edmpar, \u00e9 necess\u00e1rio que\u00a0seja \u00edmpar o n\u00famero de n\u00fameros \u00edmpares a inscrever nessas faces. H\u00e1, por isso, duas alternativas:<\/p>\n<p>A: Um n\u00famero \u00edmpar: que pode ser escolhido de ${}^{3}{{C}_{1}}=3$ maneiras diferentes;<\/p>\n<p>B: Tr\u00eas n\u00fameros \u00edmpares:\u00a0que podem ser escolhidos de ${}^{3}{{C}_{3}}=1$ maneira diferente.<\/p>\n<p>Relativamente \u00e0 primeira alternativa, cada uma das possibilidades de escolher apenas um dos tr\u00eas n\u00fameros \u00edmpares \u00e9 acompanhada pelos restantes 3 n\u00fameros pares, para serem inscritos nas quatro faces dispon\u00edveis do cubo, sobrando dois n\u00fameros \u00edmpares para inscrever nas faces dispon\u00edveis da pir\u00e2mide.<br \/>\nEsta alternativa, configura ${{N}_{A}}=({}^{3}{{C}_{1}}\\times {}^{3}{{C}_{3}}\\times {{P}_{4}})\\times {{P}_{2}}=3\\times 1\\times 24\\times 2=144$ modos diferentes de inscrever os restantes seis n\u00fameros nas faces do poliedro.<\/p>\n<p>Relativamente \u00e0 segunda alternativa, a \u00fanica possibilidade de escolher tr\u00eas dos tr\u00eas n\u00fameros \u00edmpares \u00e9 acompanhada por 1 dos restantes 3 n\u00fameros pares, que podem ser escolhidos de tr\u00eas modos diferentes, para serem inscritos nas quatro faces dispon\u00edveis do cubo, sobrando dois n\u00fameros pares para inscrever nas faces dispon\u00edveis da pir\u00e2mide.<br \/>\nEsta alternativa, configura ${{N}_{B}}=({}^{3}{{C}_{3}}\\times {}^{3}{{C}_{1}}\\times {{P}_{4}})\\times {{P}_{2}}=1\\times 3\\times 24\\times 2=144$ modos diferentes de inscrever os restantes seis n\u00fameros nas faces do poliedro.<\/p>\n<p>Portanto, o valor procurado \u00e9 $N={{N}_{A}}+{{N}_{B}}=288$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7226' onClick='GTTabs_show(0,7226)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um poliedro com nove faces, que pode ser decomposto num cubo e numa pir\u00e2mide quadrangular regular. Pretende-se numerar as nove faces do poliedro com os n\u00fameros de&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21048,"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-7226","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":3676,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag177-59_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7226","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=7226"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7226\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21048"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7226"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7226"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7226"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}