{"id":1494,"date":"2025-07-25T02:13:22","date_gmt":"2025-07-25T02:13:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1494"},"modified":"2026-03-24T07:09:20","modified_gmt":"2026-03-24T07:09:20","slug":"counting-principles-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles-fresh-take\/","title":{"raw":"Counting Principles: Fresh Take","rendered":"Counting Principles: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve counting problems using the Addition and Multiplication Principles.<\/li>\r\n \t<li>Solve counting problems using permutations and combinations.<\/li>\r\n \t<li>Solve counting problems using permutations involving n non-distinct objects.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using the Addition and Multiplication Principles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Addition Principle:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when counting mutually exclusive events<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If event [latex]A[\/latex] can occur in [latex]m[\/latex] ways and event [latex]B[\/latex] can occur in [latex]n[\/latex] ways, and [latex]A[\/latex] and [latex]B[\/latex] cannot occur simultaneously, then [latex]A[\/latex] OR [latex]B[\/latex] can occur in [latex]m + n[\/latex] ways<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication Principle:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when counting sequential events or choices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If event [latex]A[\/latex] can occur in [latex]m[\/latex] ways and event [latex]B[\/latex] can occur in [latex]n[\/latex] ways after [latex]A[\/latex] has occurred, then [latex]A[\/latex] AND [latex]B[\/latex] can occur in [latex]m \\times n[\/latex] ways<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Also known as the Fundamental Counting Principle<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Customizable product options<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Menu combinations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Outfit selections<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Password possibilities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">And many more real-world scenarios<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">A local library is organizing its annual summer reading program. They have different reading lists for different age groups and interests. The program is structured as follows:<\/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\">Children's Section (ages 5-12):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]20[\/latex] picture books<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]15[\/latex] early chapter books<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]10[\/latex] middle-grade novels<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Teen Section (ages 13-17):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]25[\/latex] young adult novels<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]12[\/latex] graphic novels<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]8[\/latex] non-fiction books<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adult Section (ages 18+):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]30[\/latex] fiction novels<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]22[\/latex] non-fiction books<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]18[\/latex] biographies<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]14[\/latex] poetry collections<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Each participant must choose one book from their age group's list to read and review. Additionally, the library decides to allow teens to also choose from the Adult Section if they prefer.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>How many total book options are there across all age groups?<\/li>\r\n \t<li>If a family with a 7-year-old, a 15-year-old, and a 40-year-old all participate, how many different combinations of book choices could they make as a family?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"929441\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"929441\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Total book options across all age groups:\r\nChildren's: [latex]20 + 15 + 10 = 45[\/latex]\r\nTeen: [latex]25 + 12 + 8 = 45[\/latex]\r\nAdult: [latex]30 + 22 + 18 + 14 = 84[\/latex]\r\nTotal: [latex]45 + 45 + 84 = 174[\/latex] books<\/li>\r\n \t<li>Family combinations:\r\n7-year-old: [latex]45[\/latex] choices\r\n15-year-old: [latex]45[\/latex] choices\r\n40-year-old: [latex]84[\/latex] choices\r\nTotal combinations: [latex]45 \\times 45 \\times 84 = 170,100[\/latex] books<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">A new phone app allows users to create a custom avatar. They can choose:<\/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]1[\/latex] of [latex]8[\/latex] face shapes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1[\/latex] of [latex]6[\/latex] hairstyles<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Any number of accessories from a set of [latex]5[\/latex] (including choosing no accessories)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1[\/latex] of [latex]10[\/latex] shirt colors<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">How many unique avatars can be created?<\/p>\r\n[reveal-answer q=\"192731\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"192731\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Face shapes: [latex]8[\/latex] choices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Hairstyles: [latex]6[\/latex] choices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Accessories: For each accessory, we have [latex]2 [\/latex]choices (use it or not). So for [latex]5[\/latex] accessories: [latex]2^5 = 32[\/latex] choices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shirt colors: [latex]10[\/latex] choices<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Total number of unique avatars = [latex]8 \\times 6 \\times 32 \\times 10 = 15,360[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]15,360[\/latex] unique avatars can be created.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are [latex]3[\/latex] types of breakfast sandwiches, [latex]4[\/latex] side dish options, and [latex]5[\/latex] beverage choices. Find the total number of possible breakfast specials.[reveal-answer q=\"559695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"559695\"]There are [latex]60[\/latex] possible breakfast specials.[\/hidden-answer]<\/section>\r\n<h2>Finding the Number of Permutations of [latex]n[\/latex] Distinct Objects<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Permutation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">An ordered arrangement of objects<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Deals with the question \"How many ways can we arrange n distinct objects?\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Permutation Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: total number of objects<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: number of objects being arranged<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Also written as [latex]_nP_r[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special Case:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">When arranging all [latex]n[\/latex] objects: [latex]P(n,n) = n![\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication Principle Connection:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Permutations are an application of the Multiplication Principle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n \\cdot (n-1) \\cdot (n-2) \\cdot ... \\cdot (n-r+1)[\/latex] factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A family of five is having portraits taken. Use the Multiplication Principle to find the following. How many ways can the family line up for the portrait?[reveal-answer q=\"847892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"847892\"][latex]120[\/latex][\/hidden-answer]How many ways can the photographer line up [latex]3[\/latex] family members?[reveal-answer q=\"416580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"416580\"][latex]60[\/latex][\/hidden-answer]How many ways can the family line up for the portrait if the parents are required to stand on each end?[reveal-answer q=\"518944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518944\"][latex]12[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">A play has a cast of [latex]7[\/latex] actors preparing to make their curtain call. Use the permutation formula to find the following.How many ways can the [latex]7[\/latex] actors line up?[reveal-answer q=\"288859\"]Show Solution[\/reveal-answer][hidden-answer a=\"288859\"][latex]P\\left(7,7\\right)=5\\text{,}040[\/latex][\/hidden-answer]How many ways can [latex]5[\/latex] of the [latex]7[\/latex] actors be chosen to line up?[reveal-answer q=\"754095\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"754095\"][latex]P\\left(7,5\\right)=2\\text{,}520[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-afgedffe-HpbuBGrHuGQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HpbuBGrHuGQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-afgedffe-HpbuBGrHuGQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851409&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-afgedffe-HpbuBGrHuGQ&amp;vembed=0&amp;video_id=HpbuBGrHuGQ&amp;video_target=tpm-plugin-afgedffe-HpbuBGrHuGQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Possible+Number+of+4+Color+Striped+Flags+(Permutation)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Possible Number of 4 Color Striped Flags (Permutation)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Finding the Number of Permutations of [latex]n[\/latex] Non-Distinct Objects<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Permutations where some objects are identical or indistinguishable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: For [latex]n[\/latex] objects, where [latex]r_1[\/latex] are alike, [latex]r_2[\/latex] are alike, etc., up to [latex]r_k[\/latex]: [latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot ... \\cdot r_k!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reasoning:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Start with total permutations ([latex]n![\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide by permutations of each set of identical objects<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This removes duplicate counts<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the number of rearrangements of the letters in the word CARRIER.[reveal-answer q=\"688005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688005\"][latex]840[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">A florist is creating a large display using:<\/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]8[\/latex] red roses<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]6[\/latex] white lilies<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]5[\/latex] yellow daisies<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]4[\/latex] purple orchids<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">How many different linear arrangements of these flowers are possible?<\/p>\r\n[reveal-answer q=\"564419\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"564419\"]\r\n<p class=\"whitespace-pre-wrap break-words\">Total flowers: [latex]8 + 6 + 5 + 4 = 23[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Apply the formula: [latex]\\frac{23!}{8! \\cdot 6! \\cdot 5! \\cdot 4!}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Using a calculator (due to large numbers): = [latex]1,185,851,393,600[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">There are over [latex]1.18[\/latex] trillion possible arrangements!<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-affefghb-c5o7G9rdLCE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/c5o7G9rdLCE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-affefghb-c5o7G9rdLCE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851410&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-affefghb-c5o7G9rdLCE&amp;vembed=0&amp;video_id=c5o7G9rdLCE&amp;video_target=tpm-plugin-affefghb-c5o7G9rdLCE\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Determine+the+Number+of+Permutations+With+Repeated+Items_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine the Number of Permutations With Repeated Items\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Combinations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Combination:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A selection of objects where order doesn't matter<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation: [latex]C(n,r)[\/latex] or [latex]_nC_r[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combination Formula: [latex]C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: total number of objects<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: number of objects being selected<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relation to Permutations: [latex]C(n,r) = \\frac{P(n,r)}{r!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Property: [latex]C(n,r) = C(n,n-r)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">An ice cream shop offers [latex]10[\/latex] flavors of ice cream. How many ways are there to choose [latex]3[\/latex] flavors for a banana split?[reveal-answer q=\"536738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"536738\"][latex]C\\left(10,3\\right)=120[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">A book club has [latex]15[\/latex] members. They need to form three committees: a [latex]5[\/latex]-person event planning committee, a [latex]4[\/latex]-person book selection committee, and a [latex]3[\/latex]-person finance committee. How many ways can they form these committees if each person can only be on one committee?[reveal-answer q=\"579868\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"579868\"]\r\n<p class=\"whitespace-pre-wrap break-words\">This is a multi-step combination problem:<\/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\">Choose [latex]5[\/latex] for event planning: [latex]C(15,5)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">From remaining [latex]10[\/latex], choose [latex]4[\/latex] for book selection: [latex]C(10,4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">From remaining [latex]6[\/latex], choose [latex]3[\/latex] for finance: [latex]C(6,3)[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Total ways = [latex]C(15,5) \\cdot C(10,4) \\cdot C(6,3)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]= \\frac{15!}{5!10!} \\cdot \\frac{10!}{4!6!} \\cdot \\frac{6!}{3!3!}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]= 3003 \\cdot 210 \\cdot 20 = 12,612,600[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">There are [latex]12,612,600[\/latex] ways to form the committees.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhegbghb-_JGn7KXG4ug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_JGn7KXG4ug?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhegbghb-_JGn7KXG4ug\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851411&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhegbghb-_JGn7KXG4ug&amp;vembed=0&amp;video_id=_JGn7KXG4ug&amp;video_target=tpm-plugin-hhegbghb-_JGn7KXG4ug\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Number+of+Ways+3+Varieties+can+be+Selected+from+12.+(Combination)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Number of Ways 3 Varieties can be Selected from 12. (Combination)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahcegcfc-UbJ-hj4Zw7k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UbJ-hj4Zw7k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahcegcfc-UbJ-hj4Zw7k\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851412&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahcegcfc-UbJ-hj4Zw7k&amp;vembed=0&amp;video_id=UbJ-hj4Zw7k&amp;video_target=tpm-plugin-ahcegcfc-UbJ-hj4Zw7k\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Evaluate+a+Combination+and+a+Permutation+-+(n%2C1)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate a Combination and a Permutation - (n,1)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Finding the Number of Subsets of a Set<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Subset:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A collection of elements from a set, including the empty set and the set itself<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Power Set:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The set of all possible subsets of a given set<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For a set with [latex]n[\/latex] elements, the number of subsets is [latex]2^n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Connection to Combinations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Number of subsets = [latex]\\sum_{k=0}^n C(n,k) = 2^n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A sundae bar at a wedding has [latex]6[\/latex] toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?[reveal-answer q=\"719477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719477\"][latex]64[\/latex] sundaes[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve counting problems using the Addition and Multiplication Principles.<\/li>\n<li>Solve counting problems using permutations and combinations.<\/li>\n<li>Solve counting problems using permutations involving n non-distinct objects.<\/li>\n<\/ul>\n<\/section>\n<h2>Using the Addition and Multiplication Principles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Addition Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when counting mutually exclusive events<\/li>\n<li class=\"whitespace-normal break-words\">If event [latex]A[\/latex] can occur in [latex]m[\/latex] ways and event [latex]B[\/latex] can occur in [latex]n[\/latex] ways, and [latex]A[\/latex] and [latex]B[\/latex] cannot occur simultaneously, then [latex]A[\/latex] OR [latex]B[\/latex] can occur in [latex]m + n[\/latex] ways<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when counting sequential events or choices<\/li>\n<li class=\"whitespace-normal break-words\">If event [latex]A[\/latex] can occur in [latex]m[\/latex] ways and event [latex]B[\/latex] can occur in [latex]n[\/latex] ways after [latex]A[\/latex] has occurred, then [latex]A[\/latex] AND [latex]B[\/latex] can occur in [latex]m \\times n[\/latex] ways<\/li>\n<li class=\"whitespace-normal break-words\">Also known as the Fundamental Counting Principle<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Customizable product options<\/li>\n<li class=\"whitespace-normal break-words\">Menu combinations<\/li>\n<li class=\"whitespace-normal break-words\">Outfit selections<\/li>\n<li class=\"whitespace-normal break-words\">Password possibilities<\/li>\n<li class=\"whitespace-normal break-words\">And many more real-world scenarios<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">A local library is organizing its annual summer reading program. They have different reading lists for different age groups and interests. The program is structured as follows:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Children&#8217;s Section (ages 5-12):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]20[\/latex] picture books<\/li>\n<li class=\"whitespace-normal break-words\">[latex]15[\/latex] early chapter books<\/li>\n<li class=\"whitespace-normal break-words\">[latex]10[\/latex] middle-grade novels<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Teen Section (ages 13-17):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]25[\/latex] young adult novels<\/li>\n<li class=\"whitespace-normal break-words\">[latex]12[\/latex] graphic novels<\/li>\n<li class=\"whitespace-normal break-words\">[latex]8[\/latex] non-fiction books<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Adult Section (ages 18+):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]30[\/latex] fiction novels<\/li>\n<li class=\"whitespace-normal break-words\">[latex]22[\/latex] non-fiction books<\/li>\n<li class=\"whitespace-normal break-words\">[latex]18[\/latex] biographies<\/li>\n<li class=\"whitespace-normal break-words\">[latex]14[\/latex] poetry collections<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Each participant must choose one book from their age group&#8217;s list to read and review. Additionally, the library decides to allow teens to also choose from the Adult Section if they prefer.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>How many total book options are there across all age groups?<\/li>\n<li>If a family with a 7-year-old, a 15-year-old, and a 40-year-old all participate, how many different combinations of book choices could they make as a family?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q929441\">Show Answer<\/button><\/p>\n<div id=\"q929441\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Total book options across all age groups:<br \/>\nChildren&#8217;s: [latex]20 + 15 + 10 = 45[\/latex]<br \/>\nTeen: [latex]25 + 12 + 8 = 45[\/latex]<br \/>\nAdult: [latex]30 + 22 + 18 + 14 = 84[\/latex]<br \/>\nTotal: [latex]45 + 45 + 84 = 174[\/latex] books<\/li>\n<li>Family combinations:<br \/>\n7-year-old: [latex]45[\/latex] choices<br \/>\n15-year-old: [latex]45[\/latex] choices<br \/>\n40-year-old: [latex]84[\/latex] choices<br \/>\nTotal combinations: [latex]45 \\times 45 \\times 84 = 170,100[\/latex] books<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">A new phone app allows users to create a custom avatar. They can choose:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]1[\/latex] of [latex]8[\/latex] face shapes<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1[\/latex] of [latex]6[\/latex] hairstyles<\/li>\n<li class=\"whitespace-normal break-words\">Any number of accessories from a set of [latex]5[\/latex] (including choosing no accessories)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1[\/latex] of [latex]10[\/latex] shirt colors<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">How many unique avatars can be created?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q192731\">Show Answer<\/button><\/p>\n<div id=\"q192731\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Face shapes: [latex]8[\/latex] choices<\/li>\n<li class=\"whitespace-normal break-words\">Hairstyles: [latex]6[\/latex] choices<\/li>\n<li class=\"whitespace-normal break-words\">Accessories: For each accessory, we have [latex]2[\/latex]choices (use it or not). So for [latex]5[\/latex] accessories: [latex]2^5 = 32[\/latex] choices<\/li>\n<li class=\"whitespace-normal break-words\">Shirt colors: [latex]10[\/latex] choices<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Total number of unique avatars = [latex]8 \\times 6 \\times 32 \\times 10 = 15,360[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]15,360[\/latex] unique avatars can be created.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are [latex]3[\/latex] types of breakfast sandwiches, [latex]4[\/latex] side dish options, and [latex]5[\/latex] beverage choices. Find the total number of possible breakfast specials.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q559695\">Show Solution<\/button><\/p>\n<div id=\"q559695\" class=\"hidden-answer\" style=\"display: none\">There are [latex]60[\/latex] possible breakfast specials.<\/div>\n<\/div>\n<\/section>\n<h2>Finding the Number of Permutations of [latex]n[\/latex] Distinct Objects<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Permutation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">An ordered arrangement of objects<\/li>\n<li class=\"whitespace-normal break-words\">Deals with the question &#8220;How many ways can we arrange n distinct objects?&#8221;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Permutation Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: total number of objects<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: number of objects being arranged<\/li>\n<li class=\"whitespace-normal break-words\">Also written as [latex]_nP_r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Special Case:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When arranging all [latex]n[\/latex] objects: [latex]P(n,n) = n![\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication Principle Connection:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Permutations are an application of the Multiplication Principle<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n \\cdot (n-1) \\cdot (n-2) \\cdot ... \\cdot (n-r+1)[\/latex] factors<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A family of five is having portraits taken. Use the Multiplication Principle to find the following. How many ways can the family line up for the portrait?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q847892\">Show Solution<\/button><\/p>\n<div id=\"q847892\" class=\"hidden-answer\" style=\"display: none\">[latex]120[\/latex]<\/div>\n<\/div>\n<p>How many ways can the photographer line up [latex]3[\/latex] family members?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q416580\">Show Solution<\/button><\/p>\n<div id=\"q416580\" class=\"hidden-answer\" style=\"display: none\">[latex]60[\/latex]<\/div>\n<\/div>\n<p>How many ways can the family line up for the portrait if the parents are required to stand on each end?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q518944\">Show Solution<\/button><\/p>\n<div id=\"q518944\" class=\"hidden-answer\" style=\"display: none\">[latex]12[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A play has a cast of [latex]7[\/latex] actors preparing to make their curtain call. Use the permutation formula to find the following.How many ways can the [latex]7[\/latex] actors line up?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q288859\">Show Solution<\/button><\/p>\n<div id=\"q288859\" class=\"hidden-answer\" style=\"display: none\">[latex]P\\left(7,7\\right)=5\\text{,}040[\/latex]<\/div>\n<\/div>\n<p>How many ways can [latex]5[\/latex] of the [latex]7[\/latex] actors be chosen to line up?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q754095\">Show Solution<\/button><\/p>\n<div id=\"q754095\" class=\"hidden-answer\" style=\"display: none\">[latex]P\\left(7,5\\right)=2\\text{,}520[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-afgedffe-HpbuBGrHuGQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HpbuBGrHuGQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-afgedffe-HpbuBGrHuGQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851409&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-afgedffe-HpbuBGrHuGQ&amp;vembed=0&amp;video_id=HpbuBGrHuGQ&amp;video_target=tpm-plugin-afgedffe-HpbuBGrHuGQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Possible+Number+of+4+Color+Striped+Flags+(Permutation)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Possible Number of 4 Color Striped Flags (Permutation)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding the Number of Permutations of [latex]n[\/latex] Non-Distinct Objects<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Permutations where some objects are identical or indistinguishable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula: For [latex]n[\/latex] objects, where [latex]r_1[\/latex] are alike, [latex]r_2[\/latex] are alike, etc., up to [latex]r_k[\/latex]: [latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot ... \\cdot r_k!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Reasoning:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with total permutations ([latex]n![\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Divide by permutations of each set of identical objects<\/li>\n<li class=\"whitespace-normal break-words\">This removes duplicate counts<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the number of rearrangements of the letters in the word CARRIER.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q688005\">Show Solution<\/button><\/p>\n<div id=\"q688005\" class=\"hidden-answer\" style=\"display: none\">[latex]840[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">A florist is creating a large display using:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]8[\/latex] red roses<\/li>\n<li class=\"whitespace-normal break-words\">[latex]6[\/latex] white lilies<\/li>\n<li class=\"whitespace-normal break-words\">[latex]5[\/latex] yellow daisies<\/li>\n<li class=\"whitespace-normal break-words\">[latex]4[\/latex] purple orchids<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">How many different linear arrangements of these flowers are possible?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q564419\">Show Answer<\/button><\/p>\n<div id=\"q564419\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Total flowers: [latex]8 + 6 + 5 + 4 = 23[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Apply the formula: [latex]\\frac{23!}{8! \\cdot 6! \\cdot 5! \\cdot 4!}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Using a calculator (due to large numbers): = [latex]1,185,851,393,600[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">There are over [latex]1.18[\/latex] trillion possible arrangements!<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-affefghb-c5o7G9rdLCE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/c5o7G9rdLCE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-affefghb-c5o7G9rdLCE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851410&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-affefghb-c5o7G9rdLCE&amp;vembed=0&amp;video_id=c5o7G9rdLCE&amp;video_target=tpm-plugin-affefghb-c5o7G9rdLCE\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Determine+the+Number+of+Permutations+With+Repeated+Items_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine the Number of Permutations With Repeated Items\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Combinations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Combination:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A selection of objects where order doesn&#8217;t matter<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]C(n,r)[\/latex] or [latex]_nC_r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combination Formula: [latex]C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: total number of objects<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: number of objects being selected<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relation to Permutations: [latex]C(n,r) = \\frac{P(n,r)}{r!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Property: [latex]C(n,r) = C(n,n-r)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">An ice cream shop offers [latex]10[\/latex] flavors of ice cream. How many ways are there to choose [latex]3[\/latex] flavors for a banana split?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q536738\">Show Solution<\/button><\/p>\n<div id=\"q536738\" class=\"hidden-answer\" style=\"display: none\">[latex]C\\left(10,3\\right)=120[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A book club has [latex]15[\/latex] members. They need to form three committees: a [latex]5[\/latex]-person event planning committee, a [latex]4[\/latex]-person book selection committee, and a [latex]3[\/latex]-person finance committee. How many ways can they form these committees if each person can only be on one committee?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q579868\">Show Answer<\/button><\/p>\n<div id=\"q579868\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">This is a multi-step combination problem:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose [latex]5[\/latex] for event planning: [latex]C(15,5)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">From remaining [latex]10[\/latex], choose [latex]4[\/latex] for book selection: [latex]C(10,4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">From remaining [latex]6[\/latex], choose [latex]3[\/latex] for finance: [latex]C(6,3)[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Total ways = [latex]C(15,5) \\cdot C(10,4) \\cdot C(6,3)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]= \\frac{15!}{5!10!} \\cdot \\frac{10!}{4!6!} \\cdot \\frac{6!}{3!3!}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]= 3003 \\cdot 210 \\cdot 20 = 12,612,600[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">There are [latex]12,612,600[\/latex] ways to form the committees.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhegbghb-_JGn7KXG4ug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_JGn7KXG4ug?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhegbghb-_JGn7KXG4ug\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851411&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhegbghb-_JGn7KXG4ug&amp;vembed=0&amp;video_id=_JGn7KXG4ug&amp;video_target=tpm-plugin-hhegbghb-_JGn7KXG4ug\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Number+of+Ways+3+Varieties+can+be+Selected+from+12.+(Combination)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Number of Ways 3 Varieties can be Selected from 12. (Combination)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahcegcfc-UbJ-hj4Zw7k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UbJ-hj4Zw7k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahcegcfc-UbJ-hj4Zw7k\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851412&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahcegcfc-UbJ-hj4Zw7k&amp;vembed=0&amp;video_id=UbJ-hj4Zw7k&amp;video_target=tpm-plugin-ahcegcfc-UbJ-hj4Zw7k\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Evaluate+a+Combination+and+a+Permutation+-+(n%2C1)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate a Combination and a Permutation &#8211; (n,1)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding the Number of Subsets of a Set<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Subset:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A collection of elements from a set, including the empty set and the set itself<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Power Set:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The set of all possible subsets of a given set<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a set with [latex]n[\/latex] elements, the number of subsets is [latex]2^n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Connection to Combinations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Number of subsets = [latex]\\sum_{k=0}^n C(n,k) = 2^n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A sundae bar at a wedding has [latex]6[\/latex] toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q719477\">Show Solution<\/button><\/p>\n<div id=\"q719477\" class=\"hidden-answer\" style=\"display: none\">[latex]64[\/latex] sundaes<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Determine the Possible Number of 4 Color Striped Flags (Permutation)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/HpbuBGrHuGQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Determine the Number of Permutations With Repeated Items\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/c5o7G9rdLCE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Determine the Number of Ways 3 Varieties can be Selected from 12. (Combination)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/_JGn7KXG4ug\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Evaluate a Combination and a Permutation - (n,1)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/UbJ-hj4Zw7k\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex: Determine the Possible Number of 4 Color Striped Flags (Permutation)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/HpbuBGrHuGQ","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 1: Determine the Number of Permutations With Repeated Items","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/c5o7G9rdLCE","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Determine the Number of Ways 3 Varieties can be Selected from 12. (Combination)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/_JGn7KXG4ug","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Evaluate a Combination and a Permutation - (n,1)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/UbJ-hj4Zw7k","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851409&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-afgedffe-HpbuBGrHuGQ&vembed=0&video_id=HpbuBGrHuGQ&video_target=tpm-plugin-afgedffe-HpbuBGrHuGQ'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851410&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-affefghb-c5o7G9rdLCE&vembed=0&video_id=c5o7G9rdLCE&video_target=tpm-plugin-affefghb-c5o7G9rdLCE'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851411&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhegbghb-_JGn7KXG4ug&vembed=0&video_id=_JGn7KXG4ug&video_target=tpm-plugin-hhegbghb-_JGn7KXG4ug'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851412&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahcegcfc-UbJ-hj4Zw7k&vembed=0&video_id=UbJ-hj4Zw7k&video_target=tpm-plugin-ahcegcfc-UbJ-hj4Zw7k'><\/script>\n","media_targets":["tpm-plugin-afgedffe-HpbuBGrHuGQ","tpm-plugin-affefghb-c5o7G9rdLCE","tpm-plugin-hhegbghb-_JGn7KXG4ug","tpm-plugin-ahcegcfc-UbJ-hj4Zw7k"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1494"}],"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\/1494\/revisions"}],"predecessor-version":[{"id":5980,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1494\/revisions\/5980"}],"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\/1494\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1494"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1494"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1494"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}