{"id":3208,"date":"2025-08-15T23:21:50","date_gmt":"2025-08-15T23:21:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3208"},"modified":"2026-03-26T17:29:33","modified_gmt":"2026-03-26T17:29:33","slug":"counting-principles-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles-apply-it\/","title":{"raw":"Counting Principles: Apply It","rendered":"Counting Principles: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using the Addition and Multiplication Principles.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using permutations and combinations.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using permutations involving n non-distinct objects.<\/li>\r\n<\/ul>\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\"><\/div>\r\n<\/div>\r\n<div class=\"h-8\">\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Music Festival Planning<\/h2>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A regional music festival is in the final stages of planning their three-day event. The organizing team needs to make a series of decisions \u2014 from scheduling performers to designing merchandise bundles \u2014 and counting principles will help them understand exactly how many options they're working with.<\/p>\r\n\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Scheduling the Opening Night Lineup<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The festival has confirmed <strong>8 performers<\/strong> for opening night. The production team needs to finalize the performance order for the <strong>first 3 slots<\/strong> of the evening, as these spots receive the most promotion and media coverage. The order matters \u2014 the opening act, the second act, and the headline opener are each featured differently in marketing materials.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">How many ways can the production team arrange 3 performers in the first 3 slots from their 8 confirmed acts?\r\n\r\n[reveal-answer q=\"599404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"599404\"]Since the order of the slots matters, this is a permutation problem. [latex] \\begin{align} P(8, 3) &amp;= \\frac{8!}{(8-3)!} \\[6pt] &amp;= \\frac{8!}{5!} \\[6pt] &amp;= 8 \\times 7 \\times 6 \\[6pt] &amp;= 336 \\text{ possible arrangements} \\end{align} [\/latex] There are 336 different ways the team can schedule those three featured slots.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[ohm_question hide_question_numbers=1]321998[\/ohm_question]<\/p>\r\n\r\n<\/section>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Selecting the Volunteer Leadership Team<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The festival relies on a pool of 150\u00a0trained volunteers. For the event weekend, the organizing committee needs to select 5\u00a0volunteers to serve as team leaders. All 5 leaders have the same responsibilities and title.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">How many ways can the committee select 5 team leaders from 150 volunteers?\r\n\r\n[reveal-answer q=\"466399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466399\"]\r\n\r\nSince all leaders have equal roles, order does not matter. This is a combination problem.\r\n\r\n[latex] \\begin{align} C(15, 4) &amp;= \\frac{15!}{4!(15-4)!} \\[6pt] &amp;= \\frac{15!}{4! \\cdot 11!} \\[6pt] &amp;= \\frac{15 \\times 14 \\times 13 \\times 12}{4 \\times 3 \\times 2 \\times 1} \\[6pt] &amp;= \\frac{32{,}760}{24} \\[6pt] &amp;= 1{,}365 \\text{ ways to select the leadership team} \\end{align} [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321999[\/ohm_question]\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Solve counting problems using the Addition and Multiplication Principles.<\/li>\n<li style=\"font-weight: 400;\">Solve counting problems using permutations and combinations.<\/li>\n<li style=\"font-weight: 400;\">Solve counting problems using permutations involving n non-distinct objects.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><\/div>\n<\/div>\n<div class=\"h-8\">\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Music Festival Planning<\/h2>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A regional music festival is in the final stages of planning their three-day event. The organizing team needs to make a series of decisions \u2014 from scheduling performers to designing merchandise bundles \u2014 and counting principles will help them understand exactly how many options they&#8217;re working with.<\/p>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Scheduling the Opening Night Lineup<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The festival has confirmed <strong>8 performers<\/strong> for opening night. The production team needs to finalize the performance order for the <strong>first 3 slots<\/strong> of the evening, as these spots receive the most promotion and media coverage. The order matters \u2014 the opening act, the second act, and the headline opener are each featured differently in marketing materials.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">How many ways can the production team arrange 3 performers in the first 3 slots from their 8 confirmed acts?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q599404\">Show Solution<\/button><\/p>\n<div id=\"q599404\" class=\"hidden-answer\" style=\"display: none\">Since the order of the slots matters, this is a permutation problem. [latex]\\begin{align} P(8, 3) &= \\frac{8!}{(8-3)!} \\[6pt] &= \\frac{8!}{5!} \\[6pt] &= 8 \\times 7 \\times 6 \\[6pt] &= 336 \\text{ possible arrangements} \\end{align}[\/latex] There are 336 different ways the team can schedule those three featured slots.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><iframe loading=\"lazy\" id=\"ohm321998\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321998&theme=lumen&iframe_resize_id=ohm321998&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Selecting the Volunteer Leadership Team<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The festival relies on a pool of 150\u00a0trained volunteers. For the event weekend, the organizing committee needs to select 5\u00a0volunteers to serve as team leaders. All 5 leaders have the same responsibilities and title.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">How many ways can the committee select 5 team leaders from 150 volunteers?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q466399\">Show Solution<\/button><\/p>\n<div id=\"q466399\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since all leaders have equal roles, order does not matter. This is a combination problem.<\/p>\n<p>[latex]\\begin{align} C(15, 4) &= \\frac{15!}{4!(15-4)!} \\[6pt] &= \\frac{15!}{4! \\cdot 11!} \\[6pt] &= \\frac{15 \\times 14 \\times 13 \\times 12}{4 \\times 3 \\times 2 \\times 1} \\[6pt] &= \\frac{32{,}760}{24} \\[6pt] &= 1{,}365 \\text{ ways to select the leadership team} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321999\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321999&theme=lumen&iframe_resize_id=ohm321999&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":11,"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":"apply_it","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\/3208"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3208\/revisions"}],"predecessor-version":[{"id":6052,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3208\/revisions\/6052"}],"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\/3208\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3208"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3208"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3208"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}