{"id":224160,"date":"2025-06-02T18:26:25","date_gmt":"2025-06-02T18:26:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=224160"},"modified":"2025-06-02T18:26:28","modified_gmt":"2025-06-02T18:26:28","slug":"what-is-the-permutation-of-14-taken-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/02\/what-is-the-permutation-of-14-taken-5\/","title":{"rendered":"What is the permutation of 14 taken 5"},"content":{"rendered":"\n<p>What is the permutation of 14 taken 5? 3. A group of 45 people is going to run a race. In how many possible ways can they be arranged as first, second, and third placers? 4. A group of 10 students wants to elect a president, vice president, secretary, and treasurer. How many different ways can they choose the officers? 5. How many ways can you introduce the 5 starting players of the Wildcats Basketball Team in the basketball game?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Answers<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>P(9,3)=504P(9,3)=504<\/li>\n\n\n\n<li>P(14,5)=240,240P(14,5)=240{,}240<\/li>\n\n\n\n<li>85 140 possible podium orders<\/li>\n\n\n\n<li>5 040 possible officer slates<\/li>\n\n\n\n<li>120 possible introduction orders<\/li>\n<\/ol>\n\n\n\n<p><strong>Why those numbers are correct<\/strong><\/p>\n\n\n\n<p>A <em>permutation<\/em> counts the number of ways to arrange rr distinct items chosen from a pool of nn distinct items \u2014 <em>with order mattering<\/em>. The standard symbol is P(n,r)P(n,r) or nPrnP r. The counting rule comes straight from the <strong>Fundamental Principle of Counting<\/strong> (\u201cmultiply the number of choices you have at each step\u201d).<\/p>\n\n\n\n<p>Mathematically, P(n,r)=n!(n\u2212r)!,P(n,r)=\\frac{n!}{(n-r)!},<\/p>\n\n\n\n<p>because n!n! lists every possible full ordering of nn objects, and dividing by (n\u2212r)!(n-r)! cancels the trailing positions we no longer care about.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>P(9,3)P(9,3).<\/strong><br>For the first slot you may choose any of 9 elements, then 8 remain, then 7 remain.<br>9\u00d78\u00d77=504.9\\times 8\\times 7=504.<\/li>\n\n\n\n<li><strong>P(14,5)P(14,5).<\/strong><br>Multiply the first five descending integers from 14:<br>14\u00d713\u00d712\u00d711\u00d710=240\u2009240.14\\times13\\times12\\times11\\times10=240\\,240.<\/li>\n\n\n\n<li><strong>Race podium from 45 runners.<\/strong><br>Gold, silver, and bronze are distinct, so order counts: 45P345P3.<br>45\u00d744\u00d743=85\u200914045\\times44\\times43=85\\,140 possible podium line-ups.<\/li>\n\n\n\n<li><strong>Electing four different officers from 10 students.<\/strong><br>Positions (president, vice president, secretary, treasurer) are inherently ordered, so again a permutation: 10P410P4.<br>10\u00d79\u00d78\u00d77=5\u200904010\\times9\\times8\\times7=5\\,040.<br>(Notice that if we merely <em>chose<\/em> a four-person committee with no roles, we\u2019d use combinations, but roles make order relevant.)<\/li>\n\n\n\n<li><strong>Introducing a five-player basketball line-up.<\/strong><br>All five starters must be named, and the announcer controls the sequence. That is a complete ordering of five objects: 5P5=5!=1205P5=5!=120 possible sequences.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>P(9,3)P(9,3).<\/strong><br>For the first slot you may choose any of 9 elements, then 8 remain, then 7 remain.<br>9\u00d78\u00d77=504.9\\times 8\\times 7=504.<\/li>\n\n\n\n<li><strong>P(14,5)P(14,5).<\/strong><br>Multiply the first five descending integers from 14:<br>14\u00d713\u00d712\u00d711\u00d710=240\u2009240.14\\times13\\times12\\times11\\times10=240\\,240.<\/li>\n\n\n\n<li><strong>Race podium from 45 runners.<\/strong><br>Gold, silver, and bronze are distinct, so order counts: 45P345P3.<br>45\u00d744\u00d743=85\u200914045\\times44\\times43=85\\,140 possible podium line-ups.<\/li>\n\n\n\n<li><strong>Electing four different officers from 10 students.<\/strong><br>Positions (president, vice president, secretary, treasurer) are inherently ordered, so again a permutation: 10P410P4.<br>10\u00d79\u00d78\u00d77=5\u200904010\\times9\\times8\\times7=5\\,040.<br>(Notice that if we merely <em>chose<\/em> a four-person committee with no roles, we\u2019d use combinations, but roles make order relevant.)<\/li>\n\n\n\n<li><strong>Introducing a five-player basketball line-up.<\/strong><br>All five starters must be named, and the announcer controls the sequence. That is a complete ordering of five objects: 5P5=5!=1205P5=5!=120 possible sequences.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>Key takeaway: whenever the wording distinguishes <em>first\/second\/third<\/em>, <em>roles<\/em>, or <em>sequence<\/em> in any way, you should expect a permutation, not a combination. The factorial shortcut nPr=n!\u2009\u2063\/\u2009\u2063(n\u2212r)!nP r=n!\\!\/\\!(n-r)! provides a quick, reliable path: list the first rr factors of n!n! and multiply. The examples above simply plug specific nn and rr values into that universal rule.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-161.jpeg\" alt=\"\" class=\"wp-image-224161\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>What is the permutation of 14 taken 5? 3. A group of 45 people is going to run a race. In how many possible ways can they be arranged as first, second, and third placers? 4. A group of 10 students wants to elect a president, vice president, secretary, and treasurer. How many different ways [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-224160","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/224160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=224160"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/224160\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=224160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=224160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=224160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}