{"id":4049,"date":"2021-08-15T12:49:52","date_gmt":"2021-08-15T12:49:52","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4049"},"modified":"2021-11-27T21:59:29","modified_gmt":"2021-11-27T16:29:29","slug":"permutation-combination-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/permutation-combination-formula\/","title":{"rendered":"Permutation and Combination Formula – Properties"},"content":{"rendered":"

Here you will learn formula permutation and combination and properties of permutation and combination with examples.<\/p>\n

Let’s begin –<\/p>\n

Permutation and Combination Formula<\/h2>\n

Permutation<\/h2>\n

Each of the arrangements in a definite order which can be made by taking some or all of the things at a time is called a PERMUTATION. In permutation, order of appearance of things is taken into account; when the order is changed, a different permutation is obtained.<\/p>\n

Generally, it involves the problems of arrangements (standing in a line, seated in a row), problems on digit, problems on letters from a word etc.<\/p>\n

\n

\\(^{n}P_r\\) denotes, the number of permutations of n different things, taken r at a time (n \\(\\in\\) N, r \\(\\in\\) W, r \\(\\le\\) n)<\/p>\n

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

Note :<\/strong><\/p>\n

(i) \\(^{n}P_n\\) = n!, \\(^{n}P_0\\) = 1, \\(^{n}P_1\\) = n<\/p>\n

(ii) Number of arrangements of n distinct things taken all at a time = n!.<\/p>\n

(iii) \\(^{n}P_r\\) is also denoted by P(n,r).<\/p>\n

Combination<\/h2>\n

Each of the groups or selections which can be made by taking some or all of the things without considering the order of the things in each groups is called a COMBINATION.<\/p>\n

Generally, involves the problem of selections, choosing, distributed groups formation, committee formation, geometrical problems etc.<\/p>\n

\n

\\(^{n}C_r\\) denotes the number of combinations of n different things taken r at a time (n \\(\\in\\) N, r \\(\\in\\) W, r \\(\\le\\) n)<\/p>\n

\\(^{n}C_r\\) = \\(n!\\over r!(n-r)!\\)<\/p>\n<\/blockquote>\n

Note :<\/strong><\/p>\n

(i)  \\(^{n}C_r\\) is also denoted by \\(\\binom{n}{r}\\) or C(n,r).<\/p>\n

(ii)  (ii) \\(^{n}P_r\\) = \\(^{n}C_r\\).r!<\/p>\n\n\n

Example : <\/span> How many four letter words can be formed from the letters of the word ‘ANSWER’ ? How many of these words start with vowel ?<\/p>\n

Solution : <\/span> Number of ways of arranging 4 different letters from 6 different letters are

\n\t\t       \\(^{6}C_4\\).4! = 360

\n\t\t       There are two vowels (A & E) in the word ‘ANSWER’

\n\t\t       Total number of 4 letter words starting with A : A _ _ _ = \\(^{5}C_3\\).3! = 60

\n\t\t       Total number of 4 letter words starting with E : E _ _ _ = \\(^{5}C_3\\).3! = 60

\n\t\t       \\(\\therefore\\)    Total number of 4 letter words starting with a vowel = 60 + 60 = 120

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

Properties of Permutation and Combination<\/h2>\n

(a)  The number of permutation of n different objects taken r at a time, when p particular objects are always to be included is r!.\\(^{n-p}C_{r-p}\\) (\\(p\\le r\\le n\\)).<\/p>\n

(b)  The number of permutation of n differnt objects taken r at a time, when repetition is allowed any number of times is \\(n^r\\).<\/p>\n

(c)  Following properties of \\(^{n}C_r\\) should be remembered:<\/p>\n

(i)  \\(^{n}C_r\\) = \\(^{n}C_{n-r}\\); \\(^{n}C_0\\) = \\(^{n}C_n\\) = 1<\/p>\n

(ii)  \\(^{n}C_x\\) = \\(^{n}C_y\\) \\(\\implies \\) x = y or x + y = n<\/p>\n

(iii)  \\(^{n}C_r\\) + \\(^{n}C_{r-1}\\) = \\(^{n+1}C_r\\) <\/p>\n

(iv)  \\(^{n}C_0\\) + \\(^{n}C_1\\) + \\(^{n}C_2\\) + ………. + \\(^{n}C_n\\) = \\(2^n\\)<\/p>\n

(v)  \\(^{n}C_r\\) = \\(n\\over r\\)\\(^{n-1}C_{r-1}\\)<\/p>\n

(vi)  \\(^{n}C_r\\) is maximum when r = \\(n\\over 2\\) if n is even & r = \\(n-1\\over 2\\) or r = \\(n+1\\over 2\\), if n is odd.<\/p>\n

(d)  The numbers of combinations of n different things taken r at a time,<\/p>\n

(i)  when p particular things are always to be included = \\(^{n-p}C_{r-p}\\)<\/p>\n

(ii)  when p particular things are always to be excluded = \\(^{n-p}C_r\\)<\/p>\n

(iii)  when p particular thing are always to be included and q particular things are to be excluded = \\(^{n-p-q}C_{r-p}\\)<\/p>\n\n\n

Example : <\/span>There are 6 pockets in the coat of a person. In how many ways can he put 4 pens in these pockets?<\/p>\n

Solution : <\/span>First pen can be put in 6 ways.

\n Similarly each of second, third and fourth pen can be put in 6 ways

\n Hence total number of ways = \\(6\\times 6\\times 6\\times 6\\) = 1296.

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

\n
Next – Total Number of Combinations <\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Addition Principle of Counting | Multiplication Principle<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula permutation and combination and properties of permutation and combination with examples. Let’s begin – Permutation and Combination Formula Permutation Each of the arrangements in a definite order which can be made by taking some or all of the things at a time is called a PERMUTATION. In permutation, order of …<\/p>\n

Permutation and Combination Formula – Properties<\/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":[529,527,528],"yoast_head":"\nPermutation and Combination Formula - Properties<\/title>\n<meta name=\"description\" content=\"In this post you will learn permutation and combination formula and properties of permutation and combination with examples.\" \/>\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\/permutation-combination-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Permutation and Combination Formula - 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