{"id":294,"date":"2012-09-08T09:46:53","date_gmt":"2012-09-08T16:46:53","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=294"},"modified":"2012-09-08T11:13:52","modified_gmt":"2012-09-08T18:13:52","slug":"summing-the-multinomial-coefficients","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/09\/summing-the-multinomial-coefficients\/","title":{"rendered":"Summing the Multinomial Coefficients"},"content":{"rendered":"<p>The sum of binomial coefficients <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_f0f98945abb6a543c2c8db30ecbd22b3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\sum_{j=0}^n {n \\choose j}\" \/><\/span><script type='math\/tex'>\\sum_{j=0}^n {n \\choose j}<\/script> equals 2<sup><em>n<\/em><\/sup>, because <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_6f15a66b7aeaacc0165f4fef7caf9a1a.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"{n \\choose j}\" \/><\/span><script type='math\/tex'>{n \\choose j}<\/script> is the number of ways to pick <em>j<\/em> elements from a set of size <em>n<\/em>, and 2<sup><em>n<\/em><\/sup> is the size of the powerset, the set of all subsets, of a set of size <em>n<\/em>: the sum, over all subset sizes, of the number of ways to choose subsets of a given size, is equal to the number of subsets. You can also see this using the binomial theorem itself:<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_9fc6730287419e04308666edc6b11f4c.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"2^{n} = (1 + 1)^{n} = \\sum_{j=0}^{n} {n \\choose j} 1^{j}1^{n-j} = \\sum_{j=0}^{n} {n \\choose j}\" \/><\/span><script type='math\/tex;  mode=display'>2^{n} = (1 + 1)^{n} = \\sum_{j=0}^{n} {n \\choose j} 1^{j}1^{n-j} = \\sum_{j=0}^{n} {n \\choose j}<\/script><\/p><\/p>\n<p>But of course there's nothing special about <em>two<\/em>; it works for multinomial coefficients just the same. The sum, over all <em>m<\/em>-tuples of subset sizes, of the number of ways to split a set of size <em>n<\/em> into subsets of sizes given by the <em>m<\/em>-tuple, is equal to the number of ways to split a set of size <em>n<\/em> into <em>m<\/em> subsets (<em>viz.<\/em>, <em>m<sup>n<\/sup><\/em>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The sum of binomial coefficients equals 2n, because is the number of ways to pick j elements from a set of size n, and 2n is the size of the powerset, the set of all subsets, of a set of &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/09\/summing-the-multinomial-coefficients\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[36],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/294"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=294"}],"version-history":[{"count":8,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/294\/revisions"}],"predecessor-version":[{"id":302,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/294\/revisions\/302"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=294"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=294"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}