{"id":4027,"date":"2021-08-14T20:01:46","date_gmt":"2021-08-14T20:01:46","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4027"},"modified":"2021-11-27T22:10:26","modified_gmt":"2021-11-27T16:40:26","slug":"dearrangement-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/dearrangement-formula\/","title":{"rendered":"Dearrangement Formula | Permutations of Alike Objects"},"content":{"rendered":"

Here you will learn dearrangement formula and permutation of alike objects with example.<\/p>\n

Let’s begin –<\/p>\n

Dearrangement Formula<\/h2>\n

There are n letters and n corresponding envelopes. The number of ways in which letters can be placed in the envelopes (one letter in each envelope) so that no letter is placed in correct envelope is<\/p>\n

\n

n![1 – \\(1\\over 1!\\) + \\(1\\over 2!\\) +……..+\\({(-1)^n}\\over n!\\)]<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>A person writes letters to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelope so that all the letters are in the wrong envelopes.<\/p>\n

Solution : <\/span>The number of ways in which all the letters can be placed in wrong envelopes.\n\t\t

= 6!(1 – \\(1\\over 2\\) – \\(1\\over 6\\) + \\(1\\over 24\\) – \\(1\\over 120\\) + \\(1\\over 720\\))

\n\t\t = 720(\\(1\\over 2\\) – \\(1\\over 6\\) + \\(1\\over 24\\) – \\(1\\over 120\\) + \\(1\\over 720\\))

\n\t\t = 360 – 120 + 30 – 6 + 1 = 265.

<\/p>\n\n\n

Permutations of alike objects<\/h2>\n

Case-1 : Taken all at a time<\/strong><\/p>\n

The number of permutations of n things taken all at a time: when p of them are similar of one type, q of them are similar of second type, r of them are similar of third type and the remaining n – (p + q + r) are all different is :<\/p>\n

\n

\\(n!\\over {p! q! r!}\\).<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> In how many ways the letter of the word “ARRANGE” can be arranged without altering the relative position of vowels & consonants.<\/p>\n

Solution : <\/span> The consonants in their position can be arranged in \\(4!\\over 2!\\) = 12 ways.

\n The vowels in their position can be arranged in \\(3!\\over 2!\\) = 3 ways.

\n \\(\\therefore\\)    total number of arrangements = \\(12\\times 3\\) = 36.

<\/p>\n\n\n

Case-2 : Taken some at a time<\/strong><\/p>\n\n\n

Example : <\/span> Find the total number of 4 letter words formed using four letters from the word \n \t\t “PARALLELOPIPED”.<\/p>\n

Solution : <\/span>Given letters are PPP, LLL, AA, EE, R, O, I, D.

\n \t\t Case 1: All distinct,    No. of words = \\(^{8}C_4\\).4! = 1680

\n \t\t Case 2: 2 alike, 2 distinct,    No. of words = \\(^{4}C_1\\).\\(^{7}C_2\\).\\(4!\\over 2!\\) = 1008\n \t\t

\n \t\t Case 3: 2 alike, 2 other alike,    No. of words = \\(^{4}C_2\\).\\(4!\\over 2! 2!\\) = 36

\n \t\t Case 4: 3 alike, 1 distinct,    No. of words = \\(^{2}C_1\\).\\(^{7}C_1\\).\\(4!\\over 3!\\) = 56\n \t\t

\n \t\t \\(\\therefore\\)    Total no. of words = 2780

<\/p>\n\n\n\n

\n
Previous – What is the Formula for Circular Permutation ?<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn dearrangement formula and permutation of alike objects with example. Let’s begin – Dearrangement Formula There are n letters and n corresponding envelopes. The number of ways in which letters can be placed in the envelopes (one letter in each envelope) so that no letter is placed in correct envelope is n![1 …<\/p>\n

Dearrangement Formula | Permutations of Alike Objects<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[18],"tags":[535,529,536],"yoast_head":"\nDearrangement Formula | Permutations of Alike Objects<\/title>\n<meta name=\"description\" content=\"In this post you will learn dearrangement formula and permutation of alike objects with example.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/dearrangement-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dearrangement Formula | Permutations of Alike Objects\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn dearrangement formula and permutation of alike objects with example.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/dearrangement-formula\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-14T20:01:46+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-27T16:40:26+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Dearrangement Formula | Permutations of Alike Objects\",\"datePublished\":\"2021-08-14T20:01:46+00:00\",\"dateModified\":\"2021-11-27T16:40:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/\"},\"wordCount\":346,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"dearrangement formula\",\"permutation and combination\",\"permutations of alike objects\"],\"articleSection\":[\"Permutation & Combination\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/dearrangement-formula\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/\",\"url\":\"https:\/\/mathemerize.com\/dearrangement-formula\/\",\"name\":\"Dearrangement Formula | Permutations of Alike Objects\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-14T20:01:46+00:00\",\"dateModified\":\"2021-11-27T16:40:26+00:00\",\"description\":\"In this post you will learn dearrangement formula and permutation of alike objects with example.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/dearrangement-formula\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/dearrangement-formula\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Dearrangement Formula | Permutations of Alike Objects\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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