{"id":220830,"date":"2024-06-25T03:11:49","date_gmt":"2024-06-25T06:11:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=220830"},"modified":"2024-06-25T03:11:49","modified_gmt":"2024-06-25T06:11:49","slug":"a-problema-para-ajudar-na-escola-par-impar","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/a-problema-para-ajudar-na-escola-par-impar\/","title":{"rendered":"(A) Problema para ajudar na escola: Par, \u00edmpar"},"content":{"rendered":"<p class=\"ti3\"  style=\"text-align:center;\">Clique no bot\u00e3o abaixo para visualizar o problema.<\/p>\n<div style=\"text-align:center;margin-top:10px;\">\n<input class=\"bo1 c1 cb13\" style=\"width:150px;\" value=\"Problema\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Problema';}\" \/>  <\/p>\n<div style=\"display: none;\">\n<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:20px; margin-bottom: 10px;\">Problema<br \/>\n<span style=\"font-size:12px;color:#0000FF\"> (A partir da 2\u00aa s\u00e9rie do E. M.- <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: Dif\u00edcil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<span style=\"font-size:12px\">(<strong>UNICAMP<\/strong>, 2004) <\/span> Considere o conjunto dos d\u00edgitos [tex]\\{1, 2, 3, &#8230;, 9\\}[\/tex] e forme com eles n\u00fameros de nove algarismos distintos.<br \/>\n<strong>(a)<\/strong> Quantos desses n\u00fameros ser\u00e3o pares?<br \/>\n<strong>(b)<\/strong> Escolhendo-se ao acaso um dos n\u00fameros do item <strong>(a)<\/strong>, qual a probabilidade de que este n\u00famero tenha exatamente dois d\u00edgitos \u00edmpares juntos?<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb36 g1\" style=\"margin:30px;line-height:30px; width:90%; margin-left:auto; margin-right:auto; \">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"90\" height=\"108\" class=\"alignright size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\"style=\"line-height:20px; margin-bottom: 10px;\"> Lembrete<\/p>\n<p><span class=\"c3\" style=\"font-size: 25px;\"> &#10000; <\/span><strong>Princ\u00edpio Fundamental da Contagem, ou Princ\u00edpio Multiplicativo, para [tex]k [\/tex] eventos<\/strong>: Se<\/p>\n<ul>\n<li>um evento <strong>E1<\/strong> puder ocorrer de [tex] m_1 [\/tex] maneiras,<\/li>\n<li> um evento <strong>E2<\/strong> puder ocorrer de [tex]m_2 [\/tex] maneiras,<\/li>\n<li> [tex]\\cdots[\/tex]<\/li>\n<li>um evento <strong>Ek<\/strong> puder ocorrer de [tex]m_k [\/tex] maneiras<\/li>\n<\/ul>\n<p> e todos esses eventos forem independentes entre si (isto \u00e9, a ocorr\u00eancia de um n\u00e3o muda a quantidade de possibilidades para a ocorr\u00eancia de outro), ent\u00e3o a quantidade de maneiras em que os [tex]k[\/tex] eventos ocorrem ao mesmo tempo \u00e9<br \/>\n[tex]\\qquad \\qquad \\boxed{m_1\\times m_2 \\times \\cdots \\times m_k} \\, .[\/tex]<br \/>\n<span style=\"font-size: 10px;\"> (Se voc\u00ea n\u00e3o se lembra desse  Princ\u00edpio, seria interessante dar uma passadinha <a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=22162\" target=\"_blank\" title=\"Sala de Estudo: Princ\u00edpio Fundamental da Contagem\"> <strong>nesta Sala de Estudo<\/strong><\/a>.)<\/span>\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<strong>(a)<\/strong> Observe que os n\u00fameros pares que podemos formar com os algarismos [tex]1, 2, 3, &#8230;, 9[\/tex] s\u00e3o aqueles que terminam em [tex]2,4, 6[\/tex] ou [tex]8[\/tex]. Dessa forma, vamos dividir a nossa contagem de n\u00fameros pares de nove algarismos distintos que podemos formar com algarismos [tex]1, 2, 3, &#8230;, 9[\/tex] em dois eventos: a escolha do \u00faltimo d\u00edgito e a escolha dos outros oito d\u00edgitos. Utilizando a nota\u00e7\u00e3o do <span class=\"c3\"><strong>Lembrete<\/strong><\/span>, calcularemos [tex]m_1[\/tex] e [tex]m_2[\/tex] e a resposta deste item ser\u00e1 o produto [tex]m_1 \\times m_2[\/tex].<\/p>\n<ul>\n<li><strong> Evento 1:<\/strong> Escolha do \u00faltimo d\u00edgito<\/li>\n<p>\tPodemos escolher o \u00faltimo d\u00edgito de [tex]4[\/tex] maneiras diferentes: ou [tex]2[\/tex] ou [tex]4[\/tex] ou [tex]6[\/tex] ou [tex]8[\/tex].<br \/>\nLogo,  [tex]\\boxed{m_1=4}[\/tex].<\/p>\n<li><strong> Evento 2:<\/strong> Escolha dos demais d\u00edgitos<\/li>\n<p>Vamos dividir este evento em oito etapas independentes entre si: escolha do primeiro d\u00edgito, escolha do segundo d\u00edgito, &#8230;, escolha do oitavo d\u00edgito; e aplicar o <strong>Princ\u00edpio Fundamental da Contagem <\/strong> para calcular a quantidade de maneiras em que esses oito eventos ocorrem ao mesmo tempo obtendo, ent\u00e3o, [tex]m_2[\/tex].<br \/>\nObserve que, dos nove algarismos dispon\u00edveis, restam oito, j\u00e1 que utilizamos um deles como \u00faltimo d\u00edgito. Podemos ent\u00e3o escolher o primeiro d\u00edgito de [tex]8[\/tex] maneiras diferentes; o segundo, de [tex]7[\/tex] maneiras; o terceiro de [tex]6[\/tex] e assim sucessivamente at\u00e9 o oitavo d\u00edgito.<br \/>\nO esqueminha abaixo ajuda na visualiza\u00e7\u00e3o da contagem.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob1.png\" style=\"border:none\" alt=\"\" width=\"1036\" height=\"71\" class=\"aligncenter size-full wp-image-170945\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob1.png 1036w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob1-300x21.png 300w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob1-768x53.png 768w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob1-1024x70.png 1024w\" sizes=\"auto, (max-width: 1036px) 100vw, 1036px\" \/><\/p>\n<p>Logo, pelo <strong>Princ\u00edpio Fundamental da Contagem<\/strong>, segue que:<br \/>\n[tex]\\qquad m_2=8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 =8![\/tex],<br \/>\nou seja,  [tex]\\boxed{m_2=40\\,320}[\/tex].<\/ul>\n<p>Portanto, podemos formar [tex] 4 \\times 40\\,320 =\\, \\fcolorbox{black}{#eee0e5}{$\\, 161\\,280$}\\,[\/tex] n\u00fameros pares com nove algarismos distintos utilizando  o conjunto dos d\u00edgitos [tex]\\{1, 2, 3, &#8230;, 9\\}[\/tex].<br \/>\n<strong>Observa\u00e7\u00e3o:<\/strong> Se voc\u00ea sabe o que \u00e9 uma Permuta\u00e7\u00e3o, voc\u00ea poderia ter obtido o valor de [tex]m_2[\/tex] rapidamente, j\u00e1 que a escolha dos oito primeiros d\u00edgitos dos n\u00fameros pares que queremos contar \u00e9, simplesmente, o n\u00famero de permuta\u00e7\u00f5es de oito objetos distintos, isto \u00e9, [tex]m_2=8![\/tex].<br \/>\nDessa forma, igualmente, o resultado final seria [tex]m_1 \\times m_2=4 \\times 8!= 161\\,280[\/tex].<\/p>\n<hr style=\"height:2px; width:100%;background-color:#800000; margin-top: 20px; margin-bottom: 5px;border: 0 none\"\/>\n<hr style=\"height:2px; width:100%;background-color:#800000; margin-top: 5px; margin-bottom: 20px;border: 0 none\"\/>\n<strong>(b)<\/strong> A escolha de d\u00edgitos do conjunto [tex]\\{1, 2, 3, &#8230;, 9\\}[\/tex] para se formar com eles n\u00fameros pares de nove algarismos distintos \u00e9 um experimento aleat\u00f3rio cujo espa\u00e7o amostral, embora com muitos elementos, \u00e9 finito e equiprov\u00e1vel. Assim, escolhendo-se ao acaso um dos n\u00fameros do item <strong>(a)<\/strong>, a probabilidade [tex]P[\/tex] de que este n\u00famero tenha exatamente dois d\u00edgitos \u00edmpares juntos \u00e9 dada pela raz\u00e3o entre &quot;casos favor\u00e1veis&quot; e &quot; casos poss\u00edveis &quot;, ou seja:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/06\/prob430_1.png\" style=\"border:none\" alt=\"\" width=\"500\" height=\"97\" class=\"aligncenter size-full wp-image-156305\" \/><\/p>\n<p>Para que um n\u00famero par com nove d\u00edgitos distintos e todos os nove diferentes de [tex]0[\/tex] tenha exatamente dois algarismos \u00edmpares em posi\u00e7\u00f5es consecutivas, esse n\u00famero deve ter uma das seguintes formas:<br \/>\n[tex]\\qquad \\qquad \\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}} \\quad ; \\quad \\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}} \\quad ; \\quad  \\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}} \\quad ; \\quad \\boxed{\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}}.[\/tex]<br \/>\nVamos fazer a contagem do total de n\u00fameros com essas quatro formas separadamente.<\/p>\n<ul>\n<li><strong> Forma [tex]\\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}}[\/tex]:<\/strong> <\/li>\n<p>Temos cinco algarismos \u00edmpares para ocuparem cinco posi\u00e7\u00f5es e quatro algarismos pares para ocuparem quatro posi\u00e7\u00f5es:<\/p>\n<table class=\"bf\" style=\"border: 2px solid #ffffff; width:90%; height:auto; text-align:center; margin-left:auto; margin-right:auto; font-size:13px\">\n<tr>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \">[tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex]  <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<\/tr>\n<tr>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]5[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<\/tr>\n<\/table>\n<p>Assim, pelo <strong>Princ\u00edpio Fundamental da Contagem<\/strong><\/p>\n<ul>\n<li>os algarismos \u00edmpares poder\u00e3o ser escolhidos de [tex]5\\times 4 \\times 3 \\times 2 \\times 1=5![\/tex] modos distintos;<\/li>\n<li>os algarismos pares poder\u00e3o ser escolhidos de [tex]4 \\times 3 \\times 2 \\times 1=4![\/tex] modos distintos;<\/li>\n<\/ul>\n<p>e, portanto, temos [tex]\\boxed{5! \\times 4!}[\/tex] modos de formar esses tipos de n\u00fameros.<\/ul>\n<p>As pr\u00f3ximas contagens ser\u00e3o id\u00eanticas a esta que acabamos de fazer, mas vamos explicit\u00e1-las para um melhor entendimento!<\/p>\n<p style=\"margin:10px\">\n<ul>\n<li><strong> Forma [tex]\\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}}[\/tex]:<\/strong> <\/li>\n<p>Temos cinco algarismos \u00edmpares para ocuparem cinco posi\u00e7\u00f5es e quatro algarismos pares para ocuparem quatro posi\u00e7\u00f5es:<\/p>\n<table class=\"bf\" style=\"border: 2px solid #ffffff; width:90%; height:auto; text-align:center; margin-left:auto; margin-right:auto; font-size:13px\">\n<tr>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \">[tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex]  <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<\/tr>\n<tr>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]5[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<\/tr>\n<\/table>\n<p>Assim, pelo <strong>Princ\u00edpio Fundamental da Contagem <\/strong> temos  [tex]\\boxed{5! \\times 4!}[\/tex] modos de formar esse tipo de n\u00fameros.<\/ul>\n<p style=\"margin:10px\">\n<ul>\n<li><strong> Forma [tex]\\boxed{\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}}[\/tex]:<\/strong> <\/li>\n<p>Temos cinco algarismos \u00edmpares para ocuparem cinco posi\u00e7\u00f5es e quatro algarismos pares para ocuparem quatro posi\u00e7\u00f5es:<\/p>\n<table class=\"bf\" style=\"border: 2px solid #ffffff; width:90%; height:auto; text-align:center; margin-left:auto; margin-right:auto; font-size:13px\">\n<tr>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \">[tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex]  <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<\/tr>\n<tr>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]5[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<\/tr>\n<\/table>\n<p>Novamente, pelo <strong>Princ\u00edpio Fundamental da Contagem<\/strong> temos [tex]\\boxed{5! \\times 4!}[\/tex] modos de formar esse tipo de n\u00fameros.<\/ul>\n<p style=\"margin:10px\">\n<ul>\n<li><strong> Forma [tex]\\boxed{\\textcolor{red}{i}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}\\textcolor{red}{i}\\textcolor{blue}{p}}[\/tex]:<\/strong> <\/li>\n<p>Temos cinco algarismos \u00edmpares para ocuparem cinco posi\u00e7\u00f5es e quatro algarismos pares para ocuparem quatro posi\u00e7\u00f5es:<\/p>\n<table class=\"bf\" style=\"border: 2px solid #ffffff; width:90%; height:auto; text-align:center; margin-left:auto; margin-right:auto; font-size:13px\">\n<tr>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \">[tex]\\textcolor{red}{i}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{red}{i}[\/tex]  <\/td>\n<td class=\"bf\" align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td class=\"bf\" align=\"center\" style=\"border-bottom: 2px solid #000000 \"> [tex]\\textcolor{blue}{p}[\/tex] <\/td>\n<\/tr>\n<tr>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]5[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]4[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]3[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]2[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<td  align=\"center\" class=\"bf\" style=\"border: 2px solid #ffffff\"> [tex]1[\/tex] alg.<\/td>\n<td  align=\"center\" style=\"border: 2px solid #ffffff\">&nbsp;<\/td>\n<\/tr>\n<\/table>\n<p>Tamb\u00e9m pelo <strong>Princ\u00edpio Fundamental da Contagem<\/strong> temos [tex]\\boxed{5! \\times 4!}[\/tex] modos de formar esse tipo de n\u00fameros.<\/ul>\n<\/ul>\n<p>Dessa forma, temos um total de [tex]\\boxed{4 \\times 5! \\times 4!}[\/tex] n\u00fameros pares com nove d\u00edgitos distintos, todos os nove diferentes de zero, que tenham exatamente dois algarismos \u00edmpares em posi\u00e7\u00f5es consecutivas.<br \/>\nAqui, tamb\u00e9m observamos que se voc\u00ea sabe o que \u00e9 uma Permuta\u00e7\u00e3o, voc\u00ea poderia ter feito essa contagem observando que a quantidade das escolhas dos algarismos \u00edmpares corresponde ao n\u00famero de permuta\u00e7\u00f5es de cinco objetos distintos ([tex]5![\/tex])  e a quantidade das escolhas dos algarismos pares corresponde ao n\u00famero de permuta\u00e7\u00f5es de quatro objetos distintos ([tex]4![\/tex])  <\/p>\n<p style=\"margin:20px\">\n<p>Finalmente, a probabilidade [tex]P[\/tex] de que escolhido ao acaso um dos n\u00fameros do item <strong>(a)<\/strong> este n\u00famero tenha exatamente dois d\u00edgitos \u00edmpares juntos \u00e9 dada por:<br \/>\n[tex]\\qquad \\qquad P=\\dfrac{4 \\times 5! \\times 4!}{161280}=\\dfrac{11520}{161280}=\\dfrac{1}{14}\\approx 0,071.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p>Portanto, [tex] \\, \\fcolorbox{black}{#eee0e5}{$\\, P \\approx 7,1\\%$}\\,[\/tex].<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:640px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"80\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:520px;background-color: #FFF5EE; padding:10px; font-family: calibri; font-size: 13px;color: #1C1C1C; line-height: 20px; \"> Se for conveniente, voc\u00ea pode obter um arquivo PDF desta p\u00e1gina, com o problema e a solu\u00e7\u00e3o, clicando no bot\u00e3o abaixo.<br \/>\nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (<em>Chrome<\/em>, <em>Edge<\/em>, <em>Firefox<\/em>, <em>Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito <em><strong>Adobe Acrobat Reader<\/strong><\/em>.<br \/>\nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o <strong><em>Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar <a href=\"http:\/\/get.adobe.com\/br\/reader\/\" rel=\"noopener\" target=\"_blank\"><strong>AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 100px;\">\n<p style=\"text-align: center; border-style: dotted; border-color :#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #800000;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/06\/Par-\u00edmpar_c.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Par, \u00edmpar.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Clique no bot\u00e3o abaixo para visualizar o problema. Problema (A partir da 2\u00aa s\u00e9rie do E. M.- N\u00edvel de dificuldade: Dif\u00edcil) (UNICAMP, 2004) Considere o conjunto dos d\u00edgitos [tex]\\{1, 2, 3, &#8230;, 9\\}[\/tex] e forme com eles n\u00fameros de nove algarismos distintos. (a) Quantos desses n\u00fameros ser\u00e3o pares? (b) Escolhendo-se ao acaso um dos n\u00fameros &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/a-problema-para-ajudar-na-escola-par-impar\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-220830","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220830","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=220830"}],"version-history":[{"count":1,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220830\/revisions"}],"predecessor-version":[{"id":220831,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220830\/revisions\/220831"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=220830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}