{"id":263956,"date":"2025-07-21T10:22:19","date_gmt":"2025-07-21T10:22:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=263956"},"modified":"2025-07-21T10:22:21","modified_gmt":"2025-07-21T10:22:21","slug":"find-a-simplified-expression-for","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/21\/find-a-simplified-expression-for\/","title":{"rendered":"Find a simplified expression for\u00a0."},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

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

The correct answer is (A)<\/strong>.<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

To find a simplified expression for the generalized binomial coefficient (1\/4 choose n), we start with its definition:<\/p>\n\n\n\n

(r choose k) = [r * (r-1) * (r-2) * … * (r-k+1)] \/ k!<\/p>\n\n\n\n

For our specific case, r = 1\/4 and k = n. Substituting these values, we get:<\/p>\n\n\n\n

(1\/4 choose n) = [ (1\/4) * (1\/4 – 1) * (1\/4 – 2) * … * (1\/4 – n + 1) ] \/ n!<\/p>\n\n\n\n

Let’s simplify the product in the numerator. It consists of n terms. We can write each term as a fraction with a denominator of 4:<\/p>\n\n\n\n

    \n
  • 1\/4<\/li>\n\n\n\n
  • 1\/4 – 1 = (1 – 4)\/4 = -3\/4<\/li>\n\n\n\n
  • 1\/4 – 2 = (1 – 8)\/4 = -7\/4<\/li>\n\n\n\n
  • 1\/4 – 3 = (1 – 12)\/4 = -11\/4<\/li>\n\n\n\n
  • …<\/li>\n\n\n\n
  • 1\/4 – (n-1) = [1 – 4(n-1)]\/4 = (5 – 4n)\/4<\/li>\n<\/ul>\n\n\n\n

    The product in the numerator is:
    [1 * (-3) * (-7) * (-11) * … * (5 – 4n)] \/ 4^n<\/p>\n\n\n\n

    Now, let’s analyze the sign and magnitude of the numerator’s product. The first term, 1, is positive. All subsequent (n-1) terms are negative. The product of (n-1) negative numbers results in a sign of (-1)^(n-1).<\/p>\n\n\n\n

    Next, we consider the product of the absolute values of these numerators:
    |1| * |-3| * |-7| * |-11| * … * |5 – 4n| = 1 * 3 * 7 * 11 * … * (4n-5)
    (Note that for n \u2265 2, 5 – 4n is negative, so |5 – 4n| = -(5-4n) = 4n-5).<\/p>\n\n\n\n

    Combining the sign, the product of absolute values, and the denominator, we get:<\/p>\n\n\n\n

    (1\/4 choose n) = [ (-1)^(n-1) * (1 * 3 * 7 * 11 * … * (4n-5)) ] \/ (4^n * n!)<\/p>\n\n\n\n

    This expression matches option (A). The product in option (A) is written as 3 * 7 * 11 * … * (4n-5). This is mathematically equivalent to 1 * 3 * 7 * 11 * … * (4n-5), as multiplying by 1 does not change the value. For the case n=1, the product is empty, which is defined as 1, correctly giving the numerator (-1)^0 * 1 = 1.<\/p>\n\n\n\n

    Therefore, the simplified expression is:<\/p>\n\n\n\n

    (A) [(-1)^(n-1) * 3 * 7 * 11 * … * (4n-5)] \/ (4^n * n!)<\/strong><\/p>\n\n\n\n

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

    The Correct Answer and Explanation is: The correct answer is (A). Explanation To find a simplified expression for the generalized binomial coefficient (1\/4 choose n), we start with its definition: (r choose k) = [r * (r-1) * (r-2) * … * (r-k+1)] \/ k! For our specific case, r = 1\/4 and k = n. […]<\/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-263956","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263956","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=263956"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263956\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=263956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=263956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=263956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}