{"id":201413,"date":"2025-03-15T11:54:00","date_gmt":"2025-03-15T11:54:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=201413"},"modified":"2025-03-15T11:54:02","modified_gmt":"2025-03-15T11:54:02","slug":"how-many-permutations-of-the-letters-abcdefgh-contain-the-string-had","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/15\/how-many-permutations-of-the-letters-abcdefgh-contain-the-string-had\/","title":{"rendered":"How many permutations of the letters ABCDEFGH contain the string HAD"},"content":{"rendered":"\n<p>How many permutations of the letters ABCDEFGH contain the string HAD?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To find how many permutations of the letters ABCDEFGH contain the string &#8220;HAD,&#8221; we can treat the string &#8220;HAD&#8221; as a single block. By doing this, we reduce the problem to finding the number of permutations of fewer items.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Treat &#8220;HAD&#8221; as a block<\/h3>\n\n\n\n<p>Since &#8220;HAD&#8221; must appear together in the permutation, we treat it as a single &#8220;letter&#8221; or block. So instead of having 8 individual letters (A, B, C, D, E, F, G, H), we now have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The &#8220;HAD&#8221; block<\/li>\n\n\n\n<li>The remaining individual letters B, C, E, F, G.<\/li>\n<\/ul>\n\n\n\n<p>This gives us a total of 6 &#8220;objects&#8221; to permute: {HAD, B, C, E, F, G}.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Find the number of permutations of these 6 objects<\/h3>\n\n\n\n<p>The number of ways to arrange 6 objects is simply 6 factorial, denoted as 6!6!. Mathematically: 6!=6\u00d75\u00d74\u00d73\u00d72\u00d71=720.6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 720.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Accounting for the arrangement of letters within &#8220;HAD&#8221;<\/h3>\n\n\n\n<p>Within the block &#8220;HAD,&#8221; the order of the letters is fixed \u2014 &#8220;H&#8221; must come first, followed by &#8220;A,&#8221; and then &#8220;D.&#8221; Therefore, there are no additional permutations inside the block.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Conclusion<\/h3>\n\n\n\n<p>Since the block &#8220;HAD&#8221; has to appear together in every arrangement and no additional rearrangement of the letters within &#8220;HAD&#8221; is allowed, the total number of permutations where &#8220;HAD&#8221; appears as a string is simply 6!=7206! = 720.<\/p>\n\n\n\n<p>Thus, the number of permutations of the letters ABCDEFGH that contain the string &#8220;HAD&#8221; is <strong>720<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Key Points:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The string &#8220;HAD&#8221; is treated as a single block.<\/li>\n\n\n\n<li>The number of ways to arrange the 6 objects is 6!6!.<\/li>\n\n\n\n<li>The letters inside the &#8220;HAD&#8221; block are fixed and do not contribute additional permutations.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>How many permutations of the letters ABCDEFGH contain the string HAD? The correct answer and explanation is : To find how many permutations of the letters ABCDEFGH contain the string &#8220;HAD,&#8221; we can treat the string &#8220;HAD&#8221; as a single block. By doing this, we reduce the problem to finding the number of permutations of [&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-201413","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201413","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=201413"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201413\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=201413"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=201413"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=201413"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}