{"id":2696,"date":"2025-08-13T18:28:09","date_gmt":"2025-08-13T18:28:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2696"},"modified":"2025-09-25T20:58:03","modified_gmt":"2025-09-25T20:58:03","slug":"probability-and-counting-theory-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-and-counting-theory-background-youll-need-2\/","title":{"raw":"Probability and Counting Theory: Background You'll Need 2","rendered":"Probability and Counting Theory: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Expand the rows of Pascal's triangle<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Pascal's Triangle<\/h3>\r\n<p class=\"whitespace-normal break-words\">Pascal's triangle is a triangular arrangement of numbers where each row contains the coefficients for expanding binomials like [latex](a + b)^n[\/latex]. Each number in the triangle is the sum of the two numbers directly above it. Understanding Pascal's triangle helps you quickly find coefficients for binomial expansions.<\/p>\r\n\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">What is Pascal's Triangle?<\/h2>\r\n<p class=\"whitespace-normal break-words\">Pascal's triangle starts with 1 at the top and builds downward. Each row corresponds to the power of a binomial expansion.<\/p>\r\n[latex]\\begin{array}{cccccccc} &amp; &amp; &amp; &amp; 1 &amp; &amp; &amp; &amp; \\\\[6pt] &amp; &amp; &amp; 1 &amp; &amp; 1 &amp; &amp; &amp; \\\\[6pt] &amp; &amp; 1 &amp; &amp; 2 &amp; &amp; 1 &amp; &amp; \\\\[6pt] &amp; 1 &amp; &amp; 3 &amp; &amp; 3 &amp; &amp; 1 &amp; \\\\[6pt] \\cdots &amp; &amp; \\cdots &amp; &amp; \\cdots &amp; &amp; \\cdots &amp; \\end{array}[\/latex]\r\n<div class=\"relative group\/copy bg-bg-000\/50 border-0.5 border-border-400 rounded-lg\">\r\n<div>\r\n<pre class=\"code-block__code !my-0 !rounded-lg !text-sm !leading-relaxed\"><code><\/code><\/pre>\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<p class=\"whitespace-normal break-words\">The triangle is named after French mathematician Blaise Pascal, but it was known centuries earlier in China and other cultures.<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\">To find any entry in Pascal's triangle:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Start with 1's on both edges of every row<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For interior numbers, add the two numbers directly above<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each row has one more entry than its row number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Row [latex]n[\/latex] has [latex]n + 1[\/latex] entries<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"h-8\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312749[\/ohm_question]<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Expand the rows of Pascal&#8217;s triangle<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Pascal&#8217;s Triangle<\/h3>\n<p class=\"whitespace-normal break-words\">Pascal&#8217;s triangle is a triangular arrangement of numbers where each row contains the coefficients for expanding binomials like [latex](a + b)^n[\/latex]. Each number in the triangle is the sum of the two numbers directly above it. Understanding Pascal&#8217;s triangle helps you quickly find coefficients for binomial expansions.<\/p>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">What is Pascal&#8217;s Triangle?<\/h2>\n<p class=\"whitespace-normal break-words\">Pascal&#8217;s triangle starts with 1 at the top and builds downward. Each row corresponds to the power of a binomial expansion.<\/p>\n<p>[latex]\\begin{array}{cccccccc} & & & & 1 & & & & \\\\[6pt] & & & 1 & & 1 & & & \\\\[6pt] & & 1 & & 2 & & 1 & & \\\\[6pt] & 1 & & 3 & & 3 & & 1 & \\\\[6pt] \\cdots & & \\cdots & & \\cdots & & \\cdots & \\end{array}[\/latex]<\/p>\n<div class=\"relative group\/copy bg-bg-000\/50 border-0.5 border-border-400 rounded-lg\">\n<div>\n<pre class=\"code-block__code !my-0 !rounded-lg !text-sm !leading-relaxed\"><code><\/code><\/pre>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<p class=\"whitespace-normal break-words\">The triangle is named after French mathematician Blaise Pascal, but it was known centuries earlier in China and other cultures.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\">To find any entry in Pascal&#8217;s triangle:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Start with 1&#8217;s on both edges of every row<\/li>\n<li class=\"whitespace-normal break-words\">For interior numbers, add the two numbers directly above<\/li>\n<li class=\"whitespace-normal break-words\">Each row has one more entry than its row number<\/li>\n<li class=\"whitespace-normal break-words\">Row [latex]n[\/latex] has [latex]n + 1[\/latex] entries<\/li>\n<\/ol>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"h-8\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312749\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312749&theme=lumen&iframe_resize_id=ohm312749&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2696"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2696\/revisions"}],"predecessor-version":[{"id":4440,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2696\/revisions\/4440"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2696\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2696"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2696"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2696"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}