{"id":223832,"date":"2025-06-02T11:57:49","date_gmt":"2025-06-02T11:57:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=223832"},"modified":"2025-06-02T11:57:51","modified_gmt":"2025-06-02T11:57:51","slug":"counting-score","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/02\/counting-score\/","title":{"rendered":"Counting Score"},"content":{"rendered":"\n

Counting Score: 0\/10 0\/10 answered Question 2 Find the value of
990 40,320 39,916,800 165<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct answer: 990<\/strong><\/p>\n\n\n\n

\n\n\n\n

When you see a symbol like 11P3^{11}P_{3}, read it as \u201cthe number of permutations<\/em> of 11 objects taken 3 at a time.\u201d A permutation<\/strong> is an ordered arrangement, so switching the order of the selected items gives a different outcome. That is why we multiply rather than simply count combinations.<\/p>\n\n\n\n

Formula<\/h3>\n\n\n\n

For any non-negative integers nn and rr where r\u2264nr \\le n, nPr\u2005\u200a=\u2005\u200an!(n\u2212r)!^{n}P_{r} \\;=\\; \\frac{n!}{(n-r)!}<\/p>\n\n\n\n

The factorial symbol \u201c!\u201d means \u201cmultiply all positive integers down to 1.\u201d
So n!=n\u00d7(n\u22121)\u00d7(n\u22122)\u00d7\u22ef\u00d71n! = n \\times (n-1) \\times (n-2) \\times \\dots \\times 1.<\/p>\n\n\n\n

Apply it to 11P3^{11}P_{3}<\/h3>\n\n\n\n
    \n
  1. Plug in the numbers<\/strong>: n=11n = 11 and r=3r = 3. 11P3\u2005\u200a=\u2005\u200a11!(11\u22123)!\u2005\u200a=\u2005\u200a11!8!^{11}P_{3} \\;=\\; \\frac{11!}{(11-3)!} \\;=\\; \\frac{11!}{8!}<\/li>\n\n\n\n
  2. Cancel the common part<\/strong>:
    Notice that 11!=11\u00d710\u00d79\u00d78!11! = 11 \\times 10 \\times 9 \\times 8!.
    The 8!8! terms in numerator and denominator cancel: 11\u00d710\u00d79\u00d78!8!=11\u00d710\u00d79\\frac{11 \\times 10 \\times 9 \\times \\cancel{8!}}{\\cancel{8!}} = 11 \\times 10 \\times 9<\/li>\n\n\n\n
  3. Multiply what\u2019s left<\/strong>: 11\u00d710=110,110\u00d79=99011 \\times 10 = 110,\\quad 110 \\times 9 = 990<\/li>\n<\/ol>\n\n\n\n

    Therefore, 11P3=990^{11}P_{3} = 990.<\/p>\n\n\n\n

    What it means<\/h3>\n\n\n\n

    Imagine you have 11 distinct books and you want to arrange any 3 of them on a display shelf where position matters (first spot, second spot, third spot). There are 990 different ways to do that. If order did not<\/strong> matter\u2014say you were merely choosing 3 books to take on a trip\u2014then you\u2019d use a combination 11C3^{11}C_{3} instead, which is smaller because repeated orders are merged into a single outcome.<\/p>\n\n\n\n

    Quick mental check<\/h3>\n\n\n\n

    A rough sanity check helps avoid arithmetic slips:<\/p>\n\n\n\n

      \n
    • The number of ordered triples must be less than 11\u00d711\u00d711=1,33111 \\times 11 \\times 11 = 1{,}331 (all possible triples with<\/strong> repeats), because permutations forbid repeats.<\/li>\n\n\n\n
    • 990 lies comfortably below 1,331, while 40,320 and 39,916,800 are clearly far too large.<\/li>\n\n\n\n
    • 165 is more plausible but is actually the value of 11C3^{11}C_{3}, the combination counterpart.<\/li>\n<\/ul>\n\n\n\n

      That confirms 990 is both mathematically and intuitively consistent.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Counting Score: 0\/10 0\/10 answered Question 2 Find the value of990 40,320 39,916,800 165 The Correct Answer and Explanation is: Correct answer: 990 When you see a symbol like 11P3^{11}P_{3}, read it as \u201cthe number of permutations of 11 objects taken 3 at a time.\u201d A permutation is an ordered arrangement, so switching the order […]<\/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-223832","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223832","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=223832"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223832\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=223832"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=223832"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=223832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}