{"id":4024,"date":"2021-08-14T20:02:34","date_gmt":"2021-08-14T20:02:34","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4024"},"modified":"2022-01-16T16:55:25","modified_gmt":"2022-01-16T11:25:25","slug":"total-number-of-combinations","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/total-number-of-combinations\/","title":{"rendered":"Total Number of Combinations | Permutation and Combination"},"content":{"rendered":"

Here you will learn how to find total number of combinations and distribution of distinct objects and alike objects.<\/p>\n

Let’s begin –<\/p>\n

Total Number of Combinations<\/h2>\n

(a)  Given n different objects, the number of ways of selecting at least one of them is,<\/p>\n

\n

\\(^{n}C_1\\) + \\(^{n}C_2\\) + \\(^{n}C_3\\) +…….+ \\(^{n}C_n\\)<\/p>\n<\/blockquote>\n

This can also be stated as the total number of combinations of n distinct things.<\/p>\n

(b) (i)  Total number of ways in which it is possible to make a selection by taking some or all out of p + q + r +…….things, where p are alike of one kind, q alike of a second kind, r alike of third kind and so on is given by : <\/p>\n

\n

(p + 1)(q + 1)(r + 1)……….-1<\/p>\n<\/blockquote>\n

(ii)  The total number of ways of selecting one or more things from p identical things of one kind, q identical things of second kind, r identical things of third kind and n different things is given by:<\/p>\n

\n

(p + 1)(q + 1)(r + 1)\\(2^n\\) – 1<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> There are 3 books of mathematics, 4 of science and 5 of english. How many different collections can be made such that each collection consists of-
\n (i) one book of each subject?
\n \t\t (ii) at least one book of each subject?
\n \t\t (iii) at least one book of english?<\/p>\n

Solution : <\/span>(i) \\(^{3}C_1\\times ^{4}C_1\\times ^{5}C_1\\) = 60

\n \t\t (ii) \\((2^3 – 1)(2^4 – 1)(2^5 -1)\\) = \\(7\\times 15\\times 31\\) = 3255

\n \t\t (iii) \\((2^5 – 1)(2^3)(2^4)\\) = \\(31\\times 128\\) = 3968

<\/p>\n\n\n

Distribution of Distinct Objects and Alike Objects<\/h2>\n

(a)  Distribution of distinct objects<\/strong><\/p>\n

Number of ways in which n distinct things can be distributed to p persons if there is no restriction to the number of things recieved by them is given by :<\/p>\n

\n

\\(p^n\\)<\/p>\n<\/blockquote>\n

(b)  Distribution of alike objects<\/strong><\/p>\n

Number of ways to distribute n alike things among p persons so that each may get none, one or more thing(s) is given by<\/p>\n

\n

\\(^{n+p-1}C_{p-1}\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>In how many ways can 5 different mangoes, 4 different oranges & 3 different apples be distributed among 3 children such that each gets at least one mango ?<\/p>\n

Solution : <\/span>5 different mangoes can be distributed by following ways among 3 children such that each gets at least 1 :

\n Total number of ways : (\\(5!\\over 3! 1! 1! 2!\\) + \\(5!\\over 2! 2! 2!\\))\\(\\times 3!\\)

\n\t\t Now, the number of ways of distributing remaining fruits (i.e. 4 oranges + 3 apples) among 3 children =\n\t\t \\(3^7\\) (as each fruit has three options).\n\t\t

\n\t\t \\(\\therefore\\)    Total no. of ways = (\\(5!\\over 3! 2!\\) + \\(5!\\over {(2!)^3}\\))\\(\\times 3!\\times 3^7\\)

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

\n
Next – What is the Formula for Circular Permutation ?<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Permutation and Combination Formula \u2013 Properties<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find total number of combinations and distribution of distinct objects and alike objects. Let’s begin – Total Number of Combinations (a)  Given n different objects, the number of ways of selecting at least one of them is, \\(^{n}C_1\\) + \\(^{n}C_2\\) + \\(^{n}C_3\\) +…….+ \\(^{n}C_n\\) This can also be stated …<\/p>\n

Total Number of Combinations | Permutation and Combination<\/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":[534,533,529,532],"yoast_head":"\nTotal Number of Combinations | Permutation and Combination<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find total number of combinations and distribution of distinct objects and alike objects.\" \/>\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\/total-number-of-combinations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Total Number of Combinations | Permutation and Combination\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find total number of combinations and distribution of distinct objects and alike objects.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/total-number-of-combinations\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-14T20:02:34+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-01-16T11:25:25+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\/total-number-of-combinations\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Total Number of Combinations | Permutation and Combination\",\"datePublished\":\"2021-08-14T20:02:34+00:00\",\"dateModified\":\"2022-01-16T11:25:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/\"},\"wordCount\":413,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"distribution of alike objects\",\"distribution of distinct objects\",\"permutation and combination\",\"total number of combinations\"],\"articleSection\":[\"Permutation & Combination\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/total-number-of-combinations\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/\",\"url\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/\",\"name\":\"Total Number of Combinations | Permutation and Combination\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-14T20:02:34+00:00\",\"dateModified\":\"2022-01-16T11:25:25+00:00\",\"description\":\"In this post you will learn how to find total number of combinations and distribution of distinct objects and alike objects.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/total-number-of-combinations\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/total-number-of-combinations\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Total Number of Combinations | Permutation and Combination\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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