{"id":11823,"date":"2013-12-03T01:23:43","date_gmt":"2013-12-03T03:23:43","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=11823"},"modified":"2023-02-28T04:52:44","modified_gmt":"2023-02-28T07:52:44","slug":"problema-exposicao-de-peixes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-exposicao-de-peixes\/","title":{"rendered":".Problema: Exposi\u00e7\u00e3o de Peixes"},"content":{"rendered":"<div style=\"border: 3px solid #800000; padding: 20px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\"><span style=\"font-family: 'Georgia'; font-size: 20px; color: #6216161;\"><strong>Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 2\u00ba ano do E. M.)<\/span><\/strong><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p style=\"text-align: justify;\"><span style=\"font-family: verdana; color: #616161; line-height: 20px;\" > Em uma exposi\u00e7\u00e3o, foi montada uma fileira de  oito aqu\u00e1rios.<br \/>\nNesses aqu\u00e1rios ser\u00e3o distribu\u00eddos quatro exemplares de peixes de esp\u00e9cies diferentes e quatro exemplares de cavalos-marinhos de esp\u00e9cies diferentes, de modo que cada aqu\u00e1rio fique com um \u00fanico animal.<br \/>\nDe quantas maneiras diferentes podem ser distribu\u00eddos os animais, se dois aqu\u00e1rios consecutivos quaisquer n\u00e3o podem conter, ambos, cavalos-marinhos?<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb36 g1n\" style=\"margin:30px;line-height:25px; width:75%; margin-left:auto; margin-right:auto; \"><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\"> Ajuda<\/p>\n<p style=\"margin:20px\">\n<p><span style=\"font-size: 22px;\"><strong> &#9999;<\/strong> <\/span><strong>Princ\u00edpio Fundamental da Contagem, ou Princ\u00edpio Multiplicativo<\/strong>: Se<\/p>\n<ul>\n<li>uma decis\u00e3o <strong>D1<\/strong> puder ser tomada de [tex] m_1 [\/tex] maneiras distintas,<\/li>\n<li> uma decis\u00e3o <strong>D2<\/strong> puder ser tomada de [tex] m_2 [\/tex] maneiras distintas,<\/li>\n<li> [tex]\\cdots[\/tex]<\/li>\n<li> uma decis\u00e3o <strong>Dk<\/strong> puder ser tomada de [tex]m_k [\/tex] maneiras distintas,<\/li>\n<li>e todas essas decis\u00f5es forem independentes entre si (isto \u00e9, a escolha de uma n\u00e3o muda a quantidade de possibilidades para a escolha de outra),<\/li>\n<\/ul>\n<p> ent\u00e3o o n\u00famero total de maneiras de tomarmos sucessivamente essas [tex]k[\/tex] decis\u00f5es \u00e9 igual ao produto<br \/>\n[tex]\\qquad \\qquad \\boxed{m_1\\times m_2 \\times \\cdots \\times m_k}\\, .[\/tex]<br \/>\n<span style=\"font-size: 12px;\"> (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 style=\"border: 3px  solid  #800000; padding: 20px;text-align: justify; font-family: verdana; color: #616161; line-height: 25px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px; font-family: 'Georgia'; font-size: 20px; color: #616161;\"><strong>Solu\u00e7\u00e3o <\/strong><\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n Chamemos cada peixe de [tex]P[\/tex] e cada  cavalo marinho de [tex]C[\/tex]. Deste modo, a disposi\u00e7\u00e3o dos animais pode ser feita de cada uma das seguintes formas:<br \/>\n&nbsp;<br \/>\n<strong>(1)<\/strong>[tex]\\,\\,\\,  P-C-P-C-P-C-P-C[\/tex];<br \/>\n&nbsp;<br \/>\n<strong>(2)<\/strong>[tex]\\,\\,\\,  C-P-C-P-C-P-C-P[\/tex];<br \/>\n&nbsp;<br \/>\n<strong>(3)<\/strong>[tex]\\,\\, \\, C-P-P-C-P-C-P-C[\/tex];<br \/>\n&nbsp;<br \/>\n<strong>(4)<\/strong>[tex]\\,\\,\\,  C-P-C-P-P-C-P-C[\/tex];<br \/>\n&nbsp;<br \/>\n<strong>(5)<\/strong>[tex]\\,\\,\\,  C-P-C-P-C-P-P-C[\/tex].<br \/>\n&nbsp;<br \/>\nComo ser\u00e3o distribu\u00eddos quatro exemplares de peixes de esp\u00e9cies diferentes e quatro exemplares de cavalos-marinhos de esp\u00e9cies diferentes, para cada um desses casos existem [tex]4\\times 3\\times 2\\times 1\\times 4\\times 3\\times 2\\times 1=576[\/tex] distribui\u00e7\u00f5es diferentes.<br \/>\nAssim, no total, ser\u00e3o [tex]5\\times576=2\\ 880[\/tex] maneiras diferentes de distribuir os animais.<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 20px; margin-bottom: 20px;border: 0 none\"\/>\n<strong>Observa\u00e7\u00e3o:<\/strong> Se voc\u00ea n\u00e3o entendeu a necessidade de multiplicarmos as cinco disposi\u00e7\u00f5es apresentadas por [tex]576[\/tex], observe que o enunciado informa que os quatro peixes e os quatro cavalos-marinhos s\u00e3o de esp\u00e9cies diferentes. S\u00f3 para ilustrar, suponha que as esp\u00e9cies distintas de peixes sejam [tex]\\textcolor{red}{P_1}-\\textcolor{#12AD2B}{P_2}-\\textcolor{blue}{P_3}-\\textcolor{#FF00FF}{P_4}[\/tex] e  as esp\u00e9cies distintas dos cavalos marinhos sejam [tex]\\textcolor{red}{C_1}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{C_4}.[\/tex] Assim, por exemplo, na primeira disposi\u00e7\u00e3o poder\u00edamos ter [tex]576[\/tex] distribui\u00e7\u00f5es dos oito animais; veja algumas dessas [tex]576[\/tex] distribui\u00e7\u00f5es:<\/p>\n<ul>\n<li>[tex]  \\textcolor{red}{P_1}-\\textcolor{red}{C_1}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{red}{P_1}-\\textcolor{#12AD2B}{C_2}-\\textcolor{#12AD2B}{P_2}-\\textcolor{red}{C_1}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{red}{P_1}-\\textcolor{blue}{C_3}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{red}{C_1}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{red}{P_1}-\\textcolor{#FF00FF}{C_4}-\\textcolor{#12AD2B}{P_2}-\\textcolor{red}{C_1}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#12AD2B}{C_2}[\/tex];<\/li>\n<li>[tex]  \\textcolor{#12AD2B}{P_2}-\\textcolor{red}{C_1}-\\textcolor{red}{P_1}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{red}{P_1}-\\textcolor{red}{C_1}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{#12AD2B}{P_2}-\\textcolor{blue}{C_3}-\\textcolor{red}{P_1}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{red}{C_1}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex]  \\textcolor{#12AD2B}{P_2}-\\textcolor{#FF00FF}{C_4}-\\textcolor{red}{P_1}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{red}{C_1}[\/tex];<\/li>\n<li>[tex] \\textcolor{blue}{P_3} -\\textcolor{red}{C_1}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{red}{P_1}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{blue}{P_3} -\\textcolor{#12AD2B}{C_2}-\\textcolor{#12AD2B}{P_2}-\\textcolor{red}{C_1}-\\textcolor{red}{P_1}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{blue}{P_3} -\\textcolor{blue}{C_3}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{red}{P_1}-\\textcolor{red}{C_1}-\\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{blue}{P_3} -\\textcolor{#FF00FF}{C_4}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{red}{P_1}-\\textcolor{blue}{C_3}-\\textcolor{#FF00FF}{P_4}-\\textcolor{red}{C_1}[\/tex];<\/li>\n<li>[tex] \\textcolor{#FF00FF}{P_4}-\\textcolor{red}{C_1}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}- \\textcolor{red}{P_1}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{#FF00FF}{P_4}-\\textcolor{#12AD2B}{C_2}-\\textcolor{#12AD2B}{P_2}-\\textcolor{red}{C_1}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}- \\textcolor{red}{P_1}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{#FF00FF}{P_4}-\\textcolor{blue}{C_3}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{red}{C_1}- \\textcolor{red}{P_1}-\\textcolor{#FF00FF}{C_4}[\/tex];<\/li>\n<li>[tex] \\textcolor{#FF00FF}{P_4}-\\textcolor{#FF00FF}{C_4}-\\textcolor{#12AD2B}{P_2}-\\textcolor{#12AD2B}{C_2}-\\textcolor{blue}{P_3}-\\textcolor{blue}{C_3}- \\textcolor{red}{P_1}-\\textcolor{red}{C_1}[\/tex].<\/li>\n<\/ul>\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>&nbsp;<\/p>\n<div style=\"border: 1px solid #616161; padding: 8px;\">\n<p style=\"text-align: justify;\"><span style=\"font-family: verdana; color: #616161;\"> Participou da discuss\u00e3o do problema o Clube<strong> Math Error. <\/strong><\/span><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 2\u00ba ano do E. M.) Em uma exposi\u00e7\u00e3o, foi montada uma fileira de oito aqu\u00e1rios. Nesses aqu\u00e1rios ser\u00e3o distribu\u00eddos quatro exemplares de peixes de esp\u00e9cies diferentes e quatro exemplares de cavalos-marinhos de esp\u00e9cies diferentes, de modo que cada aqu\u00e1rio fique com um \u00fanico animal. De quantas maneiras diferentes podem ser &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-exposicao-de-peixes\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-11823","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/11823","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=11823"}],"version-history":[{"count":26,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/11823\/revisions"}],"predecessor-version":[{"id":131581,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/11823\/revisions\/131581"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=11823"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}