{"id":157,"date":"2014-05-11T12:00:28","date_gmt":"2014-05-11T19:00:28","guid":{"rendered":"http:\/\/mostlymental.com\/?p=157"},"modified":"2014-05-12T23:16:28","modified_gmt":"2014-05-13T06:16:28","slug":"choose-your-own-mathventure","status":"publish","type":"post","link":"http:\/\/mostlymental.com\/?p=157","title":{"rendered":"Choose Your Own Mathventure"},"content":{"rendered":"<p>Hello everyone!<\/p>\n<p>Today, I'd like to talk about one of the most fundamental concepts in combinatorics: binomial coefficients.\u00a0 We define <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> (read \"<span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>\") to be the number of ways to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> things.\u00a0 So, for example, there are <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_d3d9446802a44259755d38e6d163e820.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> ways to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c81e728d9d4c2f636f067f89cc14862c.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_e4da3b7fbbce2345d7772b0674a318d5.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> blocks to color red, as shown below.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/simple_choose1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-189 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/simple_choose1-110x300.png\" alt=\"simple_choose\" width=\"110\" height=\"300\" \/><\/a><\/p>\n<p><!--more--><\/p>\n<p>&nbsp;<\/p>\n<p>Now, this is combinatorics, so most of what we're going to do today is about counting the same thing in two ways.\u00a0 We'll start with the simplest first.\u00a0 Let's show <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_b8cc2b8354c104fe0cd16013e9faa1b7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 That is, there are exactly as many ways to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> items as there are to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c95146b9d90ebf8d8fa98fbef727dca4.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of them.\u00a0 Why?\u00a0 Well, there are <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_5ac7655815b2450398416544da749b4d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> ways we could color <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c81e728d9d4c2f636f067f89cc14862c.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_e4da3b7fbbce2345d7772b0674a318d5.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> blocks red, as we saw above.\u00a0 But we could just as well choose the other <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_eccbc87e4b5ce2fe28308fd9f2a7baf3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> blocks to color blue, as below.\u00a0 So <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_d0f46cc918b93988b009fe34e01a2f9d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 By the same logic, we get that <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_b8cc2b8354c104fe0cd16013e9faa1b7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_inverse1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-188 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_inverse1-245x300.png\" alt=\"choose_inverse\" width=\"245\" height=\"300\" srcset=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_inverse1-245x300.png 245w, http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_inverse1.png 446w\" sizes=\"auto, (max-width: 245px) 100vw, 245px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Let's look at another interesting result: <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_a900c9491893da7294bf46f693193871.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Why is it true?\u00a0 Let's count all the ways to pick any number of blocks from a row of length <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> to color red.\u00a0 One way we could see this is to break up the possibilities by the number of red blocks.\u00a0 If we want <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> red blocks from our <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> total blocks, there are <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> to choose them.\u00a0 Summing over all valid <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_088cf71657e98a2811d3cc03075db13f.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> total possibilities.<\/p>\n<p>Another way to think of it is that for each block, there are two possibilities: red or white.\u00a0 The colors for each are independent, so we can multiply the possibilities together.\u00a0 There are <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> blocks, each with <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c81e728d9d4c2f636f067f89cc14862c.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> possible colorings, for a total of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_d1db0d9c696a8c056e7117dbbb4ef6db.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> possibilities.\u00a0 We've counted the same colorings with these two methods, so the counts must be equal.\u00a0 Thus, <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_a900c9491893da7294bf46f693193871.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p>&nbsp;<\/p>\n<p>It turns out binomial coefficients are also related to each other by the recurrence <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1a39344f21a0900a39860fe4f115dfb7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 We already know that <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> counts the ways to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> tiles to color red.\u00a0 So let's find a way to count the same thing to get the other side of the equation.\u00a0 In this case, we can divide the possibilities into those where the last tile is red and those where it isn't.\u00a0 If the last tile is red, then we have <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_f69fdffb82267fca1be8c6913635b318.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> more tiles to choose from, and we want <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_0e4b332cc0e37403dff51019ff361775.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> to be red.\u00a0 If it isn't, we want to choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of the remaining <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_f69fdffb82267fca1be8c6913635b318.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> tiles.\u00a0 This gives us <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_702f9ac58e4fcb2876b91305cc14c1a5.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, as we wanted.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_by_last.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-176 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_by_last-300x216.png\" alt=\"choose_by_last\" width=\"300\" height=\"216\" srcset=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_by_last-300x216.png 300w, http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/choose_by_last.png 440w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>This last relationship gives us a nice way to construct the binomial coefficients.\u00a0 We can place them in a triangle structure, as below.\u00a0 You can see <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_62a189f1132ec3b5e0f2175a09e6d0dc.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> down in the sixth row.\u00a0 From the recurrence above, we know that each number in this triangle is the sum of the two above it.\u00a0 You might have seen this pattern of numbers before.\u00a0 It's known as Pascal's triangle.\u00a0 There are a lot of pretty combinatorial and number theoretic results that come out of it.\u00a0 For today, I'll just mention that this is one way to derive it.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-201 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/pascal.png\" alt=\"pascal\" width=\"192\" height=\"198\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>One last thing before I sign off for the day.\u00a0 I've been calling the <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> binomial coefficients.\u00a0 Why do they get that name?\u00a0 It turns out that <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is the coefficient of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1f31f8c0da2e32b6acaa5b9a0e5154e9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> in <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_360d2dfdb8677162713925353a4ae3b7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 To see why it works, imagine writing out the <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> copies of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9f814753ca35a804272a30c335aae743.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> we want to multiply, as below.\u00a0 To get a copy of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1f31f8c0da2e32b6acaa5b9a0e5154e9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, we\u00a0 must choose <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_8ce4b16b22b58894aa86c421e8759df3.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of the <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7b8b965ad4bca0e41ab51de7b31363a1.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> terms to give us an <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Equivalently, we can imagine a block for each term which is colored red if we choose the <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> and blue if we choose the <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c4ca4238a0b923820dcc509a6f75849b.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 This can happen in <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> ways, so we get a total of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> copies of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1f31f8c0da2e32b6acaa5b9a0e5154e9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, which we then add together.\u00a0 Thus, the coefficient of <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1f31f8c0da2e32b6acaa5b9a0e5154e9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_faafdfefb6f6bb072b219a2597fb4cc4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/binomial.png\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-175 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/binomial-300x66.png\" alt=\"binomial\" width=\"436\" height=\"96\" srcset=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/binomial-300x66.png 300w, http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/05\/binomial.png 315w\" sizes=\"auto, (max-width: 436px) 100vw, 436px\" \/><\/a><\/p>\n<a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-facebook nolightbox\" data-provider=\"facebook\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Facebook\" href=\"https:\/\/www.facebook.com\/sharer.php?u=http%3A%2F%2Fmostlymental.com&amp;t=Choose%20Your%20Own%20Mathventure&amp;s=100&amp;p[url]=http%3A%2F%2Fmostlymental.com&amp;p[images][0]=http%3A%2F%2Fmostlymental.com%2Fwp-content%2Fuploads%2F2014%2F05%2Fsimple_choose1-110x300.png&amp;p[title]=Choose%20Your%20Own%20Mathventure\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"Facebook\" title=\"Share on Facebook\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/facebook.png\" \/><\/a><a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-twitter nolightbox\" data-provider=\"twitter\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Twitter\" href=\"https:\/\/twitter.com\/intent\/tweet?url=http%3A%2F%2Fmostlymental.com&amp;text=Hey%20check%20this%20out\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"twitter\" title=\"Share on Twitter\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/twitter.png\" \/><\/a><a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-reddit nolightbox\" data-provider=\"reddit\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Reddit\" href=\"https:\/\/www.reddit.com\/submit?url=http%3A%2F%2Fmostlymental.com&amp;title=Choose%20Your%20Own%20Mathventure\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"reddit\" title=\"Share on Reddit\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/reddit.png\" \/><\/a><a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-pinterest nolightbox\" data-provider=\"pinterest\" target=\"_blank\" rel=\"nofollow\" title=\"Pin it with Pinterest\" href=\"https:\/\/pinterest.com\/pin\/create\/button\/?url=http%3A%2F%2Fmostlymental.com&amp;media=http%3A%2F%2Fmostlymental.com%2Fwp-content%2Fuploads%2F2014%2F05%2Fsimple_choose1-110x300.png&amp;description=Choose%20Your%20Own%20Mathventure\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"pinterest\" title=\"Pin it with Pinterest\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/pinterest.png\" \/><\/a><a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-linkedin nolightbox\" data-provider=\"linkedin\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Linkedin\" href=\"https:\/\/www.linkedin.com\/shareArticle?mini=true&amp;url=http%3A%2F%2Fmostlymental.com&amp;title=Choose%20Your%20Own%20Mathventure\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"linkedin\" title=\"Share on Linkedin\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/linkedin.png\" \/><\/a><a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-mail nolightbox\" data-provider=\"mail\" rel=\"nofollow\" title=\"Share by email\" href=\"mailto:?subject=Choose%20Your%20Own%20Mathventure&amp;body=Hey%20check%20this%20out:%20http%3A%2F%2Fmostlymental.com\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"mail\" title=\"Share by email\" class=\"synved-share-image synved-social-image synved-social-image-share\" width=\"48\" height=\"48\" style=\"display: inline;width:48px;height:48px;margin: 0;padding: 0;border: none;box-shadow: none\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/social\/regular\/96x96\/mail.png\" \/><\/a><a class=\"synved-social-credit\" target=\"_blank\" rel=\"nofollow\" title=\"WordPress Social Media Feather\" href=\"http:\/\/synved.com\/wordpress-social-media-feather\/\" style=\"color:#444;text-decoration:none;font-size:8px;margin-left:5px;vertical-align:10px;white-space:nowrap\"><span>by <\/span><img loading=\"lazy\" decoding=\"async\" style=\"display: inline;margin:0;padding:0;width:16px;height:16px\" width=\"16\" height=\"16\" alt=\"feather\" src=\"http:\/\/mostlymental.com\/wp-content\/plugins\/social-media-feather\/synved-social\/image\/icon.png\" \/><\/a>","protected":false},"excerpt":{"rendered":"<p>Hello everyone! Today, I'd like to talk about one of the most fundamental concepts in combinatorics: binomial coefficients.\u00a0 We define (read \" choose \") to be the number of ways to choose of things.\u00a0 So, for example, there are ways to choose of blocks to color red, as shown below. by<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,2],"tags":[],"class_list":["post-157","post","type-post","status-publish","format-standard","hentry","category-combinatorics","category-math"],"_links":{"self":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=157"}],"version-history":[{"count":10,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":203,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/157\/revisions\/203"}],"wp:attachment":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}