{"id":62,"date":"2023-09-17T07:21:34","date_gmt":"2023-09-17T07:21:34","guid":{"rendered":"https:\/\/mathority.org\/pt\/propriedades-do-coeficiente-binomial-de-numero-combinatorio\/"},"modified":"2023-09-17T07:21:34","modified_gmt":"2023-09-17T07:21:34","slug":"propriedades-do-coeficiente-binomial-de-numero-combinatorio","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/propriedades-do-coeficiente-binomial-de-numero-combinatorio\/","title":{"rendered":"N\u00famero combinat\u00f3rio (ou coeficiente binomial)"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 um n\u00famero combinat\u00f3rio e como \u00e9 calculado (f\u00f3rmula). Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de como calcular qualquer n\u00famero combinat\u00f3rio e praticar com exerc\u00edcios resolvidos passo a passo. Tamb\u00e9m mostramos todas as propriedades e aplica\u00e7\u00f5es dos n\u00fameros combinat\u00f3rios. E por fim, aprendemos como encontrar o resultado de um n\u00famero combinat\u00f3rio diretamente com a calculadora. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-un-numero-combinatorio\"><\/span> O que \u00e9 um n\u00famero combinat\u00f3rio? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Em matem\u00e1tica, o <strong>n\u00famero combinat\u00f3rio<\/strong> , tamb\u00e9m chamado de coeficiente binomial, \u00e9 o n\u00famero de combina\u00e7\u00f5es ordin\u00e1rias (combina\u00e7\u00f5es sem repeti\u00e7\u00e3o) de grupos de k elementos que podem ser formados a partir de um conjunto de n elementos (n&gt;k).<\/p>\n<p> Um n\u00famero combinat\u00f3rio \u00e9 expresso entre par\u00eanteses da seguinte forma:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-454ae3a63c42e51753f5b4cb18a4b4b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n \\\\ k \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"31\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Por outro lado, o n\u00famero combinat\u00f3rio \u00e9 lido <em>n<\/em> sobre <em>k<\/em> . Da mesma forma, <em>n<\/em> \u00e9 chamado de numerador e <em>k<\/em> \u00e9 chamado de ordem.<\/p>\n<p> Apenas com a defini\u00e7\u00e3o de n\u00famero combinat\u00f3rio fica dif\u00edcil entender seu significado. Por\u00e9m, veremos agora como o n\u00famero combinat\u00f3rio \u00e9 determinado matematicamente, e depois nos aprofundaremos neste conceito de combinat\u00f3ria. Voc\u00ea ver\u00e1 que assim entender\u00e1 melhor. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formula-del-numero-combinatorio\"><\/span> F\u00f3rmula num\u00e9rica combinat\u00f3ria<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A f\u00f3rmula para calcular o valor de um n\u00famero combinat\u00f3rio (ou coeficiente binomial) \u00e9 a seguinte: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-nombre-combinatoire.png\" alt=\"f\u00f3rmula num\u00e9rica combinat\u00f3ria\" class=\"wp-image-1780\" width=\"184\" height=\"185\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Lembre-se que em \u00e1lgebra o ponto de exclama\u00e7\u00e3o corresponde ao fatorial de um n\u00famero. E para encontrar o fatorial de um n\u00famero, voc\u00ea precisa multiplicar todos os n\u00fameros inteiros positivos de 1 por esse n\u00famero. Por exemplo, para calcular o fatorial do n\u00famero 4 voc\u00ea deve multiplicar 1, 2, 3 e 4:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-092a0b9aa93c1211bc61b5b34844a3fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4! = 1\\cdot 2 \\cdot 3 \\cdot 4 =24\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"153\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Tamb\u00e9m \u00e9 importante saber que o fatorial de 0 \u00e9 igual a 1. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-255b29d39824041f4d334887daa9d55d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0! = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplo-de-como-calcular-un-numero-combinatorio\"><\/span> Exemplo de c\u00e1lculo de um n\u00famero combinat\u00f3rio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, determinaremos passo a passo o valor de um n\u00famero combinat\u00f3rio como exemplo, para que voc\u00ea possa ver como isso \u00e9 feito:<\/p>\n<ul>\n<li> Calcule o valor do n\u00famero combinat\u00f3rio 5 sobre 3.<\/li>\n<\/ul>\n<p> O coeficiente binomial de 5 sobre 3 corresponde \u00e0 seguinte express\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bdcbd05befcedad2d69b8388a30fa5ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"29\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Portanto, se aplicarmos a f\u00f3rmula para n\u00fameros combinat\u00f3rios, para determinar seu valor devemos realizar as seguintes opera\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d1d3925d89289554a4a5cea934b1e01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{5!}{3!(5-3)!}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"130\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Ou equivalente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9dccfbf9704a343dfff5e3b7999c3891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{5!}{3!\\cdot 2!}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"99\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Portanto, encontramos os fatoriais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00345c4e6c1e39974bfae029047f40de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{1\\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5}{(1\\cdot 2 \\cdot 3) \\cdot (1\\cdot 2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"182\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A multiplica\u00e7\u00e3o 1\u00b72\u00b73 \u00e9 repetida no numerador e no denominador, ent\u00e3o a fra\u00e7\u00e3o pode ser simplificada eliminando este fator:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-652f38c698238fcd311eb1d8c9367182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{\\cancel{1\\cdot 2 \\cdot 3} \\cdot 4 \\cdot 5}{(\\cancel{1\\cdot 2 \\cdot 3)} \\cdot (1\\cdot 2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"182\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-454d692c958dbf69b0e1118f2f21982b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{ 4 \\cdot 5}{1\\cdot 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"89\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Agora calculamos os produtos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8ac93ab4573c69daa96709c15dbb84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\cfrac{20}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"76\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E por fim, fazemos a divis\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33b8f9d85113910811bd3491bd16f5a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5 \\\\ 3 \\end{pmatrix} = \\bm{10}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"74\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Propiedades-del-numero-combinatorio\"><\/span> Propriedades do n\u00famero combinat\u00f3rio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> N\u00fameros combinat\u00f3rios, ou coeficientes binomiais, podem ser combinados de acordo com as seguintes propriedades:<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Dois <strong>n\u00fameros combinat\u00f3rios complementares<\/strong> s\u00e3o aqueles que possuem o mesmo numerador <em>n<\/em> e a soma de suas ordens \u00e9 equivalente ao referido numerador. Assim, o resultado de dois n\u00fameros combinat\u00f3rios complementares \u00e9 id\u00eantico.<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8b93e5c1a92342922857dffd5ccbc61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n \\\\ k \\end{pmatrix} = \\begin{pmatrix}n \\\\ n-k \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"123\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Essa caracter\u00edstica dos n\u00fameros combinat\u00f3rios tamb\u00e9m \u00e9 chamada de identidade de simetria.<\/p>\n<p> Por exemplo, 6 sobre 4 d\u00e1 o mesmo resultado que 6 sobre 2, porque 6-4=2. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ad5997be8c69d9c99d4b72404feea1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6 \\\\ 4 \\end{pmatrix} =\\cfrac{6!}{4!(6-4)!}  = \\cfrac{\\cancel{1\\cdot 2 \\cdot 3\\cdot 4}\\cdot 5 \\cdot 6}{(\\cancel{1\\cdot 2 \\cdot 3\\cdot 4})\\cdot (1\\cdot 2)}= \\cfrac{30}{2} = \\bm{15}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"384\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a30e46edb6ddcddccd5887cecd3d1c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6 \\\\ 2 \\end{pmatrix} =\\cfrac{6!}{2!(6-2)!} = \\cfrac{\\cancel{1\\cdot 2 \\cdot 3\\cdot 4}\\cdot 5 \\cdot 6}{(1\\cdot 2)\\cdot (\\cancel{1\\cdot 2 \\cdot 3\\cdot 4})}= \\cfrac{30}{2} = \\bm{15}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"384\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">A soma de dois n\u00fameros combinat\u00f3rios com o mesmo numerador e ordens sucessivas \u00e9 igual a outro n\u00famero combinat\u00f3rio cujo numerador equivale ao numerador das adi\u00e7\u00f5es mais 1 e cuja ordem corresponde ao maior valor das ordens das adi\u00e7\u00f5es. Em outras palavras, a seguinte condi\u00e7\u00e3o \u00e9 sempre atendida:<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d4265b42b3140a5bf8d8d8e6b74b925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ k \\end{pmatrix} + \\begin{pmatrix}n\\\\ k+1 \\end{pmatrix}= \\begin{pmatrix}n+1\\\\ k+1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"210\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Por exemplo:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-952934807b87f5983dc91ed1cdc3241a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}9\\\\ 2 \\end{pmatrix} + \\begin{pmatrix}9\\\\ 3 \\end{pmatrix}= \\begin{pmatrix}10\\\\ 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Esta propriedade tamb\u00e9m \u00e9 conhecida como regra de Pascal.<\/p>\n<p> Por outro lado, esta f\u00f3rmula tamb\u00e9m pode ser aplicada ao contr\u00e1rio para decompor um n\u00famero combinat\u00f3rio em dois n\u00fameros combinat\u00f3rios mais simples:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-19cb7f9861c61d17e79fb65e0dfee9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ k \\end{pmatrix} = \\begin{pmatrix}n-1\\\\ k-1 \\end{pmatrix}+ \\begin{pmatrix}n-1\\\\ k \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"211\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Por exemplo, o n\u00famero combinat\u00f3rio 8 sobre 4 \u00e9 igual a 7 sobre 3 mais 7 sobre 4:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d6d42f8d7c9b4ba8dec3c27029b30f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}8\\\\ 4 \\end{pmatrix} = \\begin{pmatrix}7\\\\ 3 \\end{pmatrix}+ \\begin{pmatrix}7\\\\ 4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"144\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Qualquer n\u00famero positivo maior que 1 \u00e9 igual ao pr\u00f3prio n\u00famero.<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f50693630bdb11a333371c37fa023d5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ 1 \\end{pmatrix} =n\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"68\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A raz\u00e3o para esta propriedade \u00e9 que o fatorial de um n\u00famero \u00e9 igual ao fatorial do n\u00famero anterior multiplicado pelo pr\u00f3prio n\u00famero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-538fd046daf5726856948bbed0bc1b2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n \\\\ 1 \\end{pmatrix} =\\cfrac{n!}{1!(n-1)!} = \\cfrac{n\\cdot (n-1) \\cdot (n-2)\\cdots 1}{(n-1) \\cdot (n-2)\\cdots 1}= n\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"375\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Exemplos deste tipo de n\u00fameros combinat\u00f3rios:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48a62f11350db57d4ffc7cc3b4169a4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}12\\\\ 1 \\end{pmatrix} =12 \\qquad  \\begin{pmatrix}5\\\\ 1 \\end{pmatrix} =5  \\qquad \\begin{pmatrix}9\\\\ 1 \\end{pmatrix} =9\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"294\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Qualquer n\u00famero positivo maior que 0 \u00e9 igual a um.<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00194e1abf490a12e203ed5e8797c7f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ 0 \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"65\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Na verdade, o denominador da fra\u00e7\u00e3o de tal n\u00famero combinat\u00f3rio ser\u00e1 sempre igual ao numerador da fra\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ffcbe8b8aab8ef3a6a6288df9fcd9147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ 0 \\end{pmatrix} =\\cfrac{n!}{0!(n-0)!} =\\cfrac{n!}{n!} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"206\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Exemplos de n\u00fameros combinat\u00f3rios como este:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37cec31b317b48dc0080810360941f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 0 \\end{pmatrix} =1 \\qquad  \\begin{pmatrix}2\\\\ 0 \\end{pmatrix} =1 \\qquad  \\begin{pmatrix}23\\\\ 0 \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"284\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Cada n\u00famero em si \u00e9 igual a 1.<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1eb1cfb501b54a435a581adc68c2f2f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ n \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"65\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Aqui est\u00e1 a demonstra\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ce41f441169a0f1098f900d74b03b38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ n \\end{pmatrix} =\\cfrac{n!}{n!(n-n)!} =\\cfrac{n!}{n!\\cdot 0!}= \\cfrac{n!}{n!} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"277\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Exemplos de n\u00fameros combinat\u00f3rios como este: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-341fc99ef910038cf53561a9b6680f70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}5\\\\ 5 \\end{pmatrix} =1 \\qquad  \\begin{pmatrix}37\\\\ 37 \\end{pmatrix} =1 \\qquad  \\begin{pmatrix}14\\\\ 14 \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"293\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Como-calcular-un-numero-combinatorio-con-la-calculadora\"><\/span> Como calcular um n\u00famero combinat\u00f3rio com a calculadora<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> At\u00e9 agora vimos como encontrar um n\u00famero combinat\u00f3rio de n\u00fameros mais ou menos simples, mas quando temos que operar com quantidades muito grandes \u00e9 melhor usar a calculadora para determinar o n\u00famero combinat\u00f3rio. Veremos agora como inserir um n\u00famero combinat\u00f3rio na calculadora.<\/p>\n<p> Assim, a chave usada para calcular um n\u00famero combinat\u00f3rio com a calculadora \u00e9 a <strong>chave nCr<\/strong> . E para determinar o valor do n\u00famero combinat\u00f3rio, voc\u00ea deve primeiro inserir o numerador do n\u00famero combinat\u00f3rio, em segundo lugar pressionar a tecla nCr, depois inserir a ordem do n\u00famero combinat\u00f3rio e por fim pressionar a tecla igual.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0415dd33a2b02d34f25b3ce7ed9eb5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ k \\end{pmatrix}\\quad \\color{red}\\bm{\\longrightarrow} \\quad \\color{black} n \\rightarrow \\boxed{nCr} \\rightarrow k \\rightarrow \\boxed{=}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"339\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Nas calculadoras cient\u00edficas CASIO, a tecla nCr geralmente possui um bot\u00e3o pr\u00f3prio ou fica acima do bot\u00e3o de divis\u00e3o, dependendo do modelo.<\/p>\n<p> Por exemplo, se quisermos saber qual \u00e9 o n\u00famero combinat\u00f3rio 10 sobre 6, devemos fazer a seguinte sequ\u00eancia: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2644b485893664160985bcfdbea173c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}10\\\\ 6 \\end{pmatrix}\\quad \\color{red}\\bm{\\longrightarrow} \\quad \\color{black} 10 \\rightarrow \\boxed{nCr} \\rightarrow 6 \\rightarrow \\boxed{=} \\rightarrow 210\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"407\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aplicaciones-del-numero-combinatorio\"><\/span> Aplica\u00e7\u00f5es do n\u00famero combinat\u00f3rio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Se voc\u00ea chegou at\u00e9 aqui, provavelmente j\u00e1 sabe como resolver qualquer n\u00famero combinat\u00f3rio, perfeito. Mas\u2026 para que serve o n\u00famero combinat\u00f3rio? Pois bem, veremos ent\u00e3o todas as vantagens que este tipo de opera\u00e7\u00e3o t\u00e3o especial apresenta.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Combinatoria\"><\/span> Combinat\u00f3ria<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Como vimos no topo da p\u00e1gina, o resultado de um n\u00famero combinat\u00f3rio<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72853e95c6b133307a284ae9da39d7d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}n\\\\ k \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"31\" style=\"vertical-align: -17px;\"><\/p>\n<p> representa o n\u00famero de grupos poss\u00edveis de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> elementos que podem ser formados a partir de um conjunto de um total de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> Unid.<\/p>\n<p> Portanto, alguns problemas combinat\u00f3rios podem ser resolvidos usando n\u00fameros combinat\u00f3rios (ou coeficientes binomiais). Vamos ver como fazer isso usando um exemplo:<\/p>\n<ul>\n<li> Numa turma de 30 alunos, queremos escolher um grupo de 4 alunos para realizar determinadas tarefas. Qual \u00e9 o n\u00famero total de grupos diferentes que podem ser formados?<\/li>\n<\/ul>\n<p> Neste caso, a ordem dos alunos n\u00e3o importa, o mesmo aluno n\u00e3o se repete duas vezes dentro do grupo e nem todos os alunos entram no grupo. Portanto, a f\u00f3rmula num\u00e9rica combinat\u00f3ria pode ser usada para determinar de quantas maneiras o grupo pode ser formado.<\/p>\n<p> Para isso, deve-se calcular o n\u00famero combinat\u00f3rio tendo o n\u00famero total de alunos como numerador e com o n\u00famero de alunos que formar\u00e3o o grupo como ordem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11eac1a8918444a100bca85bfbd6e701_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}30 \\\\ 4 \\end{pmatrix} =\\cfrac{30!}{4!(30-4)!} =\\cfrac{30\\cdot 29 \\cdot 28 \\cdot 27 \\cdot \\cancel{26!}}{4\\cdot 3 \\cdot 2 \\cdot 1 \\cdot \\cancel{26!}} = \\cfrac{657720}{24}=\\bm{27405}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"464\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> O n\u00famero total de combina\u00e7\u00f5es poss\u00edveis \u00e9, portanto, de 27.405 grupos.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Binomio-de-Newton\"><\/span> Bin\u00f4mio de Newton<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Outra aplica\u00e7\u00e3o dos n\u00fameros combinat\u00f3rios \u00e9 o bin\u00f4mio de Newton. O bin\u00f4mio de Newton \u00e9 um polin\u00f4mio composto por dois termos elevados juntos a um n\u00famero inteiro, ou seja, o bin\u00f4mio de Newton \u00e9 aquele polin\u00f4mio que responde \u00e0 seguinte express\u00e3o alg\u00e9brica:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7cb070eb92ff2ce3199cbbf72ab6122_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^n\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Obviamente, se o bin\u00f4mio for elevado ao quadrado, isso significa que \u00e9 uma <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\">identidade not\u00e1vel<\/a><\/span><\/strong> e, portanto, pode ser facilmente calculada com a f\u00f3rmula correspondente. Por outro lado, quando o bin\u00f4mio \u00e9 elevado a n\u00fameros grandes, o c\u00e1lculo torna-se bastante dif\u00edcil. Bem, o teorema binomial de Newton diz que esses tipos de polin\u00f4mios podem ser calculados muito facilmente a partir de n\u00fameros combinat\u00f3rios.<\/p>\n<p> Clique no link a seguir e descubra o que \u00e9 <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/formula-do-teorema-binomial-ou-binomial-de-newton-e-exercicios-resolvidos\/\">a f\u00f3rmula binomial de Newton<\/a><\/span><\/strong> e como ela \u00e9 calculada. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios resolvidos passo a passo. E finalmente, voc\u00ea descobrir\u00e1 a curiosa hist\u00f3ria deste teorema. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Triangulo-de-Tartaglia-o-de-Pascal\"><\/span> Tri\u00e2ngulo de Tartaglia (ou Pascal)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Como voc\u00ea viu ao longo deste artigo, calcular manualmente o coeficiente binomial de n\u00fameros grandes pode ser trabalhoso e complicado.<\/p>\n<p> Por outro lado, com o tri\u00e2ngulo de Tartaglia, tamb\u00e9m chamado de tri\u00e2ngulo de Pascal, todos os n\u00fameros combinat\u00f3rios podem ser facilmente determinados usando uma regra mnem\u00f4nica. Isto \u00e9 logicamente muito \u00fatil, pois economiza muito tempo durante os c\u00e1lculos.<\/p>\n<p> Para descobrir exatamente como fazer isso, veja a explica\u00e7\u00e3o <a href=\"https:\/\/mathority.org\/pt\/tartaglia-ou-triangulo-de-pascal\/\"><strong><span style=\"text-decoration: underline;\">do tri\u00e2ngulo de Tartaglia<\/span><\/strong><\/a> . Nesta p\u00e1gina vinculada voc\u00ea descobrir\u00e1 o que \u00e9 esse misterioso tri\u00e2ngulo, para que serve (tem aplica\u00e7\u00f5es surpreendentes) \ud83d\ude2e e qual a sua origem (j\u00e1 era usado h\u00e1 mais de 1000 anos). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-numeros-combinatorios\"><\/span> Exerc\u00edcios de n\u00fameros combinat\u00f3rios resolvidos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea possa praticar e compreender totalmente os conceitos explicados, deixamos v\u00e1rios exerc\u00edcios resolvidos passo a passo sobre n\u00fameros combinat\u00f3rios.<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre o n\u00famero combinat\u00f3rio 9 por 5 (sem usar calculadora). <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar o valor do n\u00famero combinat\u00f3rio 9 de 5 simplesmente aplicamos a f\u00f3rmula fatorial: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e2bb8cd51adb05d2cbc2299f179ed44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}9 \\\\ 5 \\end{pmatrix} =\\cfrac{9!}{5!(9-5)!} = \\cfrac{9\\cdot 8 \\cdot 7\\cdot 6\\cdot \\cancel{5!}}{\\cancel{5!} \\cdot (4 \\cdot 3 \\cdot 2 \\cdot 1)}= \\cfrac{3024}{24} = \\bm{126}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"381\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Qual \u00e9 o resultado da seguinte soma de dois n\u00fameros combinat\u00f3rios? (sem calculadora) <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32e3c53635693ea4e1648d7d0a12288c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}10\\\\ 6 \\end{pmatrix} + \\begin{pmatrix}10\\\\ 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"103\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Das propriedades dos n\u00fameros combinat\u00f3rios, segue-se que a soma do problema \u00e9 igual ao seguinte n\u00famero combinat\u00f3rio:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-22963c45606c26da0bcb662a5534f1b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}10\\\\ 6 \\end{pmatrix} + \\begin{pmatrix}10\\\\ 7 \\end{pmatrix}=\\begin{pmatrix}11\\\\ 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"171\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, basta calcular o n\u00famero combinat\u00f3rio 11 de 7: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d56c6d97bd8761d053d98039fc064652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}11 \\\\ 7 \\end{pmatrix} =\\cfrac{11!}{7!(11-7)!} = \\cfrac{11\\cdot 10 \\cdot 9\\cdot 8 \\cdot \\cancel{7!}}{\\cancel{7!} \\cdot (4 \\cdot 3 \\cdot 2 \\cdot 1)}= \\cfrac{7920}{24} = \\bm{330}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"403\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Determine se os seguintes n\u00fameros combinat\u00f3rios s\u00e3o iguais: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be04129d96ae292a529d8bd986bc31f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 0 \\end{pmatrix} \\qquad  \\begin{pmatrix}6\\\\ 1 \\end{pmatrix}\\qquad  \\begin{pmatrix}6\\\\ 6 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"176\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar o resultado dos tr\u00eas n\u00fameros combinat\u00f3rios, n\u00e3o \u00e9 necess\u00e1rio usar uma calculadora, mas eles podem ser facilmente encontrados gra\u00e7as \u00e0s propriedades dos n\u00fameros combinat\u00f3rios.<\/p>\n<p class=\"has-text-align-left\"> Em primeiro lugar, um n\u00famero combinat\u00f3rio de qualquer n\u00famero maior que 0 d\u00e1 1. Portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fda7c6f2f56556c6af74bde199613037_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 0 \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"64\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, qualquer n\u00famero maior que um \u00e9 igual ao pr\u00f3prio n\u00famero. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e60628beab23d09954027294de2fabf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 1 \\end{pmatrix} =6\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"65\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E por fim, qualquer n\u00famero combinat\u00f3rio formado pelo mesmo n\u00famero repetido duas vezes equivale a 1. Portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-659cabb0351db5bf4148f2b2c0f536f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 6 \\end{pmatrix} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"64\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluindo, o primeiro e o terceiro n\u00fameros combinat\u00f3rios do problema s\u00e3o iguais, por\u00e9m s\u00e3o diferentes do n\u00famero combinat\u00f3rio intermedi\u00e1rio. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-18528223523fe6672b927769a53ef7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}6\\\\ 0 \\end{pmatrix}= \\begin{pmatrix}6\\\\ 6 \\end{pmatrix}\\neq \\begin{pmatrix}6\\\\ 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 um n\u00famero combinat\u00f3rio e como \u00e9 calculado (f\u00f3rmula). Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de como calcular qualquer n\u00famero combinat\u00f3rio e praticar com exerc\u00edcios resolvidos passo a passo. Tamb\u00e9m mostramos todas as propriedades e aplica\u00e7\u00f5es dos n\u00fameros combinat\u00f3rios. E por fim, aprendemos como encontrar o resultado de um n\u00famero &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/propriedades-do-coeficiente-binomial-de-numero-combinatorio\/\"> <span class=\"screen-reader-text\">N\u00famero combinat\u00f3rio (ou coeficiente binomial)<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-62","post","type-post","status-publish","format-standard","hentry","category-binomios"],"yoast_head":"<!-- This site is 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