{"id":201,"date":"2019-12-08T04:00:17","date_gmt":"2019-12-08T04:00:17","guid":{"rendered":"http:\/\/blog.newtonchineseschool.org\/lilijia\/?p=201"},"modified":"2019-12-09T03:18:45","modified_gmt":"2019-12-09T03:18:45","slug":"aops_cp-week-12-pascals-triangle","status":"publish","type":"post","link":"https:\/\/blog.newtonchineseschool.org\/lilijia\/2019\/12\/08\/aops_cp-week-12-pascals-triangle\/","title":{"rendered":"AoPS_CP Week 12: Pascal&#8217;s Triangle"},"content":{"rendered":"\n<p>AoPS_CP Week 12: Pascal&#8217;s Triangle<\/p>\n\n\n\n<p><strong>Lesson Review:<\/strong><\/p>\n\n\n\n<p>Pascal\u2019s Triangle (also known as Yanghui Triangle in China) is a triangular array constructed by summing adjacent elements in preceding rows. It is widely used in counting and polynomial expressions. <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" width=\"247\" height=\"204\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/pascals-triangle.png\" alt=\"\" class=\"wp-image-222\" \/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.1.png\" alt=\"\" class=\"wp-image-202\" width=\"206\" height=\"320\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.1.png 308w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.1-193x300.png 193w\" sizes=\"(max-width: 206px) 100vw, 206px\" \/><\/figure><\/div>\n\n\n\n<p>The entries of Pascal\u2019s Triangle are the combinations (n\nchoose r).<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"196\" height=\"196\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.3.jpg\" alt=\"\" class=\"wp-image-204\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.3.jpg 196w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.3-150x150.jpg 150w\" sizes=\"(max-width: 196px) 100vw, 196px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"335\" height=\"196\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.4.gif\" alt=\"\" class=\"wp-image-205\" \/><\/figure>\n\n\n\n<p>The numbers of Pascal\u2019s Triangle also count the number of paths from the top of the triangle to the point where the number appears.<\/p>\n\n\n\n<p><strong>The sum of each row in Pascal&#8217;s Triangle = 2^n.<\/strong><\/p>\n\n\n\n<p><strong>Pascal\u2019s Identity:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.5.png\" alt=\"\" class=\"wp-image-208\" width=\"172\" height=\"41\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.5.png 444w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2019\/12\/12.5-300x72.png 300w\" sizes=\"(max-width: 172px) 100vw, 172px\" \/><\/figure>\n\n\n\n<p><strong>Homework:<\/strong><\/p>\n\n\n\n<ul><li>P180 Ex 12.3.1, 12.3.3<\/li><li>P184 Ex 12.4.1 (a)<\/li><li>P186 Ex 12.5.1<\/li><li>P187 Ex 12.12<\/li><li>P189 Ex 12.19, 12.20<\/li><\/ul>\n\n\n\n<p><strong>Other Resources:<\/strong><\/p>\n\n\n\n<p><a href=\"https:\/\/artofproblemsolving.com\/wiki\/index.php\/Pascal%27s_triangle\">https:\/\/artofproblemsolving.com\/wiki\/index.php\/Pascal%27s_triangle<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/www.mathsisfun.com\/pascals-triangle.html\">https:\/\/www.mathsisfun.com\/pascals-triangle.html<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/medium.com\/i-math\/top-10-secrets-of-pascals-triangle-6012ba9c5e23\">https:\/\/medium.com\/i-math\/top-10-secrets-of-pascals-triangle-6012ba9c5e23<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/www.khanacademy.org\/math\/precalculus\/x9e81a4f98389efdf:polynomials\/x9e81a4f98389efdf:binomial\/v\/pascals-triangle-binomial-theorem\">https:\/\/www.khanacademy.org\/math\/precalculus\/x9e81a4f98389efdf:polynomials\/x9e81a4f98389efdf:binomial\/v\/pascals-triangle-binomial-theorem<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/artofproblemsolving.com\/wiki\/index.php\/Pascal%27s_Identity\">https:\/\/artofproblemsolving.com\/wiki\/index.php\/Pascal%27s_Identity<\/a><\/p>\n\n\n\n<p><strong>Something Else:<\/strong><\/p>\n\n\n\n<p>*I will be back to China from 12\/10 to 12\/27. Next Sunday (12\/15)\nwe will have Mr. Xu (the teacher of AoPS: Precalculus) to teach Ch.11 Expected\nValue and Ch. 13 The Hockey Stick Identity. Thank you, Mr. Xu! <\/p>\n\n\n\n<p>There is no school on 12\/22 and 12\/29. Merry Christmas and\nHappy New Year, everyone! See you on 1\/5\/2020 for our last class before the\nfinal exam on 1\/12! <\/p>\n","protected":false},"excerpt":{"rendered":"<p>AoPS_CP Week 12: Pascal&#8217;s Triangle Lesson Review: Pascal\u2019s Triangle (also known as Yanghui Triangle in China) is a triangular array constructed by summing adjacent elements in preceding rows. It is widely used in counting and polynomial expressions. The entries of Pascal\u2019s Triangle are the combinations (n choose r). The numbers of Pascal\u2019s Triangle also count [&hellip;]<\/p>\n","protected":false},"author":157,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/201"}],"collection":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/users\/157"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/comments?post=201"}],"version-history":[{"count":4,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/201\/revisions"}],"predecessor-version":[{"id":225,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/201\/revisions\/225"}],"wp:attachment":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/media?parent=201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/categories?post=201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/tags?post=201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}