{"id":2702,"date":"2025-08-13T18:28:33","date_gmt":"2025-08-13T18:28:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2702"},"modified":"2025-09-25T20:59:19","modified_gmt":"2025-09-25T20:59:19","slug":"probability-and-counting-theory-background-youll-need-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-and-counting-theory-background-youll-need-4\/","title":{"raw":"Probability and Counting Theory: Background You'll Need 4","rendered":"Probability and Counting Theory: Background You&#8217;ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Understand what factorials are and calculate them for whole numbers<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Factorials<\/h2>\r\nFactorials are a fundamental concept in mathematics, playing a crucial role in combinatorics, probability theory, and various other mathematical fields. The factorial of a non-negative integer [latex]n[\/latex], denoted as [latex]n![\/latex], is the product of all positive integers less than or equal to [latex]n[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>factorial<\/h3>\r\n<p class=\"whitespace-pre-wrap break-words\">For any non-negative integer [latex]n[\/latex], the factorial of [latex]n[\/latex] is defined as:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]n! = n \\times (n-1) \\times (n-2) \\times \\cdots \\times 3 \\times 2 \\times 1[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Special cases:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]0! = 1[\/latex] (by definition)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1! = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Let's compute the factorials for the first few natural numbers:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]1! = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]2! = 2 \\times 1 = 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]3! = 3 \\times 2 \\times 1 = 6[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]4! = 4 \\times 3 \\times 2 \\times 1 = 24[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]5! = 5 \\times 4 \\times 3 \\times 2 \\times 1 = 120[\/latex]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Factorials grow extremely quickly. Even for relatively small values of [latex]n[\/latex], [latex]n![\/latex] becomes very large. For example, [latex]10! = 3,628,800[\/latex].<\/section><section class=\"textbox example\" aria-label=\"Example\">Compute [latex]7![\/latex].[reveal-answer q=\"448496\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"448496\"]<center>[latex]\r\n\\begin{align*}\r\n7! &amp;= 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 \\\\\r\n&amp;= 7 \\times 6 \\times 5 \\times 24 \\\\\r\n&amp;= 7 \\times 6 \\times 120 \\\\\r\n&amp;= 7 \\times 720 \\\\\r\n&amp;= 5040\r\n\\end{align*}\r\n[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24999[\/ohm2_question]<\/section>For any positive integers [latex]m[\/latex] and [latex]n[\/latex] where [latex]m &lt; n[\/latex], [latex]n![\/latex] is always divisible by [latex]m![\/latex].\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-pre-wrap break-words\">When you have a fraction with factorials, look for a matching factorial pattern in both the numerator and denominator. Just like canceling regular numbers, you can cancel the smaller factorial by dividing both top and bottom by it. For example, if you see [latex]\\frac{8!}{6!}[\/latex], you can rewrite [latex]8![\/latex] as [latex]8 \\cdot 7 \\cdot 6![\/latex] and then cancel the [latex]6![\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{8!}{6!} = \\frac{8 \\cdot 7 \\cdot 6!}{6!} = 8 \\cdot 7 = 56[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25000[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Understand what factorials are and calculate them for whole numbers<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Factorials<\/h2>\n<p>Factorials are a fundamental concept in mathematics, playing a crucial role in combinatorics, probability theory, and various other mathematical fields. The factorial of a non-negative integer [latex]n[\/latex], denoted as [latex]n![\/latex], is the product of all positive integers less than or equal to [latex]n[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>factorial<\/h3>\n<p class=\"whitespace-pre-wrap break-words\">For any non-negative integer [latex]n[\/latex], the factorial of [latex]n[\/latex] is defined as:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]n! = n \\times (n-1) \\times (n-2) \\times \\cdots \\times 3 \\times 2 \\times 1[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Special cases:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]0! = 1[\/latex] (by definition)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1! = 1[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Let&#8217;s compute the factorials for the first few natural numbers:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]1! = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]2! = 2 \\times 1 = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]3! = 3 \\times 2 \\times 1 = 6[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]4! = 4 \\times 3 \\times 2 \\times 1 = 24[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]5! = 5 \\times 4 \\times 3 \\times 2 \\times 1 = 120[\/latex]<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Factorials grow extremely quickly. Even for relatively small values of [latex]n[\/latex], [latex]n![\/latex] becomes very large. For example, [latex]10! = 3,628,800[\/latex].<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Compute [latex]7![\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q448496\">Show Answer<\/button><\/p>\n<div id=\"q448496\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align*}  7! &= 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 \\\\  &= 7 \\times 6 \\times 5 \\times 24 \\\\  &= 7 \\times 6 \\times 120 \\\\  &= 7 \\times 720 \\\\  &= 5040  \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24999\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24999&theme=lumen&iframe_resize_id=ohm24999&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>For any positive integers [latex]m[\/latex] and [latex]n[\/latex] where [latex]m < n[\/latex], [latex]n![\/latex] is always divisible by [latex]m![\/latex].\n\n\n\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-pre-wrap break-words\">When you have a fraction with factorials, look for a matching factorial pattern in both the numerator and denominator. Just like canceling regular numbers, you can cancel the smaller factorial by dividing both top and bottom by it. For example, if you see [latex]\\frac{8!}{6!}[\/latex], you can rewrite [latex]8![\/latex] as [latex]8 \\cdot 7 \\cdot 6![\/latex] and then cancel the [latex]6![\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{8!}{6!} = \\frac{8 \\cdot 7 \\cdot 6!}{6!} = 8 \\cdot 7 = 56[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25000\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25000&theme=lumen&iframe_resize_id=ohm25000&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":5,"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\/2702"}],"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":2,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2702\/revisions"}],"predecessor-version":[{"id":4442,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2702\/revisions\/4442"}],"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\/2702\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2702"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2702"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2702"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}