{"id":96280,"date":"2018-02-09T12:11:07","date_gmt":"2018-02-09T14:11:07","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=96280"},"modified":"2023-07-27T02:21:01","modified_gmt":"2023-07-27T05:21:01","slug":"problema-para-ajudar-na-escola-muitas-combinacoes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-muitas-combinacoes\/","title":{"rendered":".Problema para ajudar na escola: N\u00fameros binomiais"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=198097\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#8D918D;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:35px;\">\n<p class=\"tit\"  style=\"line-height:25px;margin:10px\">Problema <span style=\"font-size:12px; color:#0000FF\"><br \/>\n(A partir da 2\u00aa s\u00e9rie do E. M. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: M\u00e9dio<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Calcule [tex]S[\/tex], sabendo-se que<br \/>\n[tex]\\qquad \\qquad \\quad \\quad S=\\dfrac{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<p><strong>Observa\u00e7\u00e3o:<\/strong>  A nota\u00e7\u00e3o [tex]\\dbinom{n}{p}[\/tex] indica o n\u00famero binomial de [tex]n[\/tex] por [tex]p[\/tex].<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<div class=\"cb36 g1n\" style=\"margin:30px; line-height:35px; width:85%; margin-left:auto; margin-right:auto; padding:20px; \">\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=\"81\" height=\"99\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\" style=\"line-height:25px;margin:10px\"> Lembrete <\/p>\n<p style=\"margin:20px\">\n<p>Se [tex]n[\/tex] e [tex]p[\/tex] s\u00e3o n\u00fameros naturais tais que [tex]p \\leqslant n\\, [\/tex], definimos o n\u00famero binomial [tex]\\dbinom{n}{p}[\/tex] por:<br \/>\n[tex]\\qquad \\boxed{\\dbinom{n}{p}=\\dfrac{n!}{\\left(n-p\\right)!\\, p!}}\\, .[\/tex]<\/p><\/div>\n<div class=\"g1\" style=\"line-height:35px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o 1 <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>[tex]\\textcolor{#800000}{(1)}[\/tex] Calculemos o numerador.<\/p>\n<ul>[tex]\\begin{align*}\\boxed{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}&#038;=\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times\\dfrac{18!}{(18-5)!\\, 5!}\\\\<br \/>\n&#038;=\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times\\dfrac{18!}{13!\\, 5!}\\\\<br \/>\n&#038;=\\dfrac{21\\times 20\\times 19\\times 18!}{13! \\times 8\\times 7\\times 6\\times 5!}\\\\<br \/>\n&#038;=\\boxed{\\dfrac{21!}{13!\\, 8!}} \\qquad \\textcolor{#800000}{(i)}<br \/>\n\\end{align*}[\/tex]<\/ul>\n<p style=\"margin:10px\">\n<p>[tex]\\textcolor{#800000}{(2)}[\/tex] Calculemos, agora, o denominador.<\/p>\n<ul>\n<ul>\n<li>Observe que:<br \/>\n[tex]\\begin{align*}\\dbinom{18}{5}+\\dbinom{18}{12}&#038;=\\dfrac{18!}{\\left(18-5\\right)!\\, 5!}+\\dfrac{18!}{\\left(18-12\\right)!\\, 12!}\\\\<br \/>\n&#038;=\\dfrac{18!}{13!\\, 5!}+\\dfrac{18!}{6!\\, 12!}=\\dfrac{18!\\times 6+18!\\times 13}{13!\\, 6!}\\\\<br \/>\n&#038;=\\dfrac{18!\\times (6+13)}{13!\\, 6!}=\\dfrac{18!\\times 19}{13!\\, 6!}\\\\<br \/>\n&#038;=\\dfrac{19!}{13!\\, 6!}\\qquad \\textcolor{#800000}{(ii)}<br \/>\n\\end{align*}[\/tex]<\/li>\n<li>Observe tamb\u00e9m que:<br \/>\n[tex]\\begin{align*}\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}&#038;=\\left[\\dbinom{18}{5}+\\dbinom{18}{12}\\right]+\\dbinom{19}{12}\\\\<br \/>\n&#038;\\stackrel{\\textcolor{#800000}{(ii)}}{=}\\dfrac{19!}{13!\\, 6!}+\\dbinom{19}{12}=\\dfrac{19!}{13!\\, 6!}+\\dfrac{19!}{(19-12)!\\, 12!}\\\\<br \/>\n&#038;=\\dfrac{19!}{13!\\, 6!}+\\dfrac{19!}{7!\\, 12!}=\\dfrac{19!\\times 7+19!\\times 13}{13!\\, 7!}\\\\<br \/>\n&#038;=\\dfrac{19!\\times (7+13)}{13!\\, 7!}=\\dfrac{19!\\times 20}{13!\\, 7!}\\\\<br \/>\n&#038;=\\dfrac{20!}{13!\\, 7!}\\qquad \\textcolor{#800000}{(iii)}<br \/>\n\\end{align*}[\/tex]<\/li>\n<\/ul>\n<\/ul>\n<p>Portanto:<br \/>\n[tex]\\begin{align*}\\boxed{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}&#038;=\\left[\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}\\right]+\\dbinom{20}{8}\\\\<br \/>\n&#038;\\stackrel{\\textcolor{#800000}{(iii)}}{=}\\dfrac{20!}{13!\\, 7!}+\\dbinom{20}{8}=\\dfrac{20!}{13!\\, 7!}+\\dfrac{20!}{(20-8)!v8!}\\\\<br \/>\n&#038;=\\dfrac{20!}{13!\\, 7!}+\\dfrac{20!}{12!\\, 8!}=\\dfrac{20!\\times 8+20!\\times 13}{13!\\, 8!}\\\\<br \/>\n&#038;=\\dfrac{20!\\times (8+13)}{13!\\, 8!}=\\dfrac{20!\\times 21}{13!\\, 8!}\\\\<br \/>\n&#038;=\\boxed{\\dfrac{21!}{13!\\, 8!}}\\qquad \\textcolor{#800000}{(iv)}<br \/>\n\\end{align*}[\/tex]<\/p>\n<p style=\"margin:30px\">\n<p>[tex]\\textcolor{#800000}{(3)}[\/tex] Por [tex]\\textcolor{#800000}{(i)}\\, [\/tex] e [tex]\\textcolor{#800000}{(iv)}\\, [\/tex], segue que:<\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad \\qquad \\fcolorbox{black}{#eee0e5}{$S=\\dfrac{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}=\\dfrac{\\dfrac{21!}{13!\\, 8!}}{\\dfrac{21!}{13!\\, 8!}}=1$}[\/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<div class=\"cb36 g1n\" style=\"margin:30px; line-height:30px; width:85%; margin-left:auto; margin-right:auto; padding:20px; \">\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=\"81\" height=\"99\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\"  style=\"line-height:25px;margin:10px\"> Ferramentas que podem ajudar na segunda solu\u00e7\u00e3o<\/p>\n<p style=\"margin:20px\">\n<p>Duas propriedades dos n\u00fameros binomiais podem encurtar a solu\u00e7\u00e3o deste problema: <\/p>\n<p style=\"margin:20px\">\n<p><span class=\"c3\"> <strong> P1: <\/strong><\/span>  [tex]\\dbinom{n}{p}=\\dbinom{n}{n-p}.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p><span class=\"c3\"> <strong>P2: <\/strong><\/span> [tex]\\dbinom{n}{p+1}+\\dbinom{n}{p}=\\dbinom{n+1}{p+1}.[\/tex]\n<\/div>\n<div class=\"g1n\" style=\"line-height:27px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o 2 <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 30px;border: 0 none\"\/>\n<ul>\n<li><strong>Numerador:<\/strong>\n<ul>[tex]\\\\<br \/>\n\\, \\\\<br \/>\n\\begin{align*}\\boxed{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}&#038;=\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times\\dfrac{18!}{(18-5)!\\, 5!}\\\\<br \/>\n&#038;=\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times\\dfrac{18!}{13!\\, 5!}\\\\<br \/>\n&#038;=\\dfrac{21\\times 20\\times 19\\times 18!}{13! \\times 8\\times 7\\times 6\\times 5!}\\\\<br \/>\n&#038;=\\boxed{\\dfrac{21!}{13!\\, 8!}} \\qquad \\textcolor{#800000}{(i)}<br \/>\n\\end{align*}[\/tex]<\/li>\n<li><strong>Denominador:<\/strong><br \/>\n[tex]\\begin{align*}\\boxed{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}&#038;\\stackrel{\\textcolor{#800000}{(P1)}}{=}\\dbinom{18}{13}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{12}\\\\<br \/>\n&#038;=\\left[\\dbinom{18}{13}+\\dbinom{18}{12}\\right] +\\dbinom{19}{12}+\\dbinom{20}{12}\\\\<br \/>\n&#038;\\stackrel{\\textcolor{#800000}{(P2)}}{=}\\dbinom{19}{13}+\\dbinom{19}{12}+\\dbinom{20}{12}\\\\<br \/>\n&#038;=\\left[\\dbinom{19}{13}+\\dbinom{19}{12}\\right]+\\dbinom{20}{12}\\\\<br \/>\n&#038;\\stackrel{\\textcolor{#800000}{(P2)}}{=}\\dbinom{20}{13}+\\dbinom{20}{12}\\\\<br \/>\n&#038;\\stackrel{\\textcolor{#800000}{(P2)}}{=}\\boxed{\\dbinom{21}{13}} \\qquad \\textcolor{#800000}{(ii)}<br \/>\n\\end{align*}[\/tex]<\/li>\n<li>Por [tex]\\textcolor{#800000}{(i)}[\/tex] e [tex]\\textcolor{#800000}{(ii)}[\/tex], temos que:<\/li>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad \\qquad \\fcolorbox{black}{#eee0e5}{$S=\\dfrac{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}=\\dfrac{\\dfrac{21!}{13!\\, 8!}}{\\dfrac{21!}{13!\\, 8!}}=1$}[\/tex] .<\/ul>\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 style=\"margin:30px\">\n<div class=\"g3\" style=\"width: 450px;\">\nGincana de 2017 &#8211; Clubes de Matem\u00e1tica da OBMEP<br \/>\nN\u00edvel B &#8211; Quest\u00e3o Dif\u00edcil<\/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\/2019\/10\/N\u00fameros-binomiais.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ N\u00fameros binomiais.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (A partir da 2\u00aa s\u00e9rie do E. M. &#8211; N\u00edvel de dificuldade: M\u00e9dio) Calcule [tex]S[\/tex], sabendo-se que [tex]\\qquad \\qquad \\quad \\quad S=\\dfrac{\\dfrac{21}{8} \\times \\dfrac{20}{7} \\times \\dfrac{19}{6} \\times \\dbinom{18}{5}}{\\dbinom{18}{5}+\\dbinom{18}{12}+\\dbinom{19}{12}+\\dbinom{20}{8}}.[\/tex] Observa\u00e7\u00e3o: A nota\u00e7\u00e3o [tex]\\dbinom{n}{p}[\/tex] indica o n\u00famero binomial de [tex]n[\/tex] por [tex]p[\/tex]. Lembrete Se [tex]n[\/tex] e [tex]p[\/tex] s\u00e3o &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-muitas-combinacoes\/\">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-96280","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/96280","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=96280"}],"version-history":[{"count":19,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/96280\/revisions"}],"predecessor-version":[{"id":198099,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/96280\/revisions\/198099"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=96280"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}