- Factorial Notation:Let
*n*be a positive integer. Then, factorial*n*, denoted*n*! is defined as:n! = n(n - 1)(n - 2) ... 3.2.1.Examples:- We define 0! = 1.
- 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.

- Permutations:The different arrangements of a given number of things by taking some or all at a time, are called permutations.Examples:
- All permutations (or arrangements) made with the letters
*a*,*b*,*c*by taking two at a time are (*ab*,*ba*,*ac*,*ca*,*bc*,*cb*). - All permutations made with the letters
*a*,*b*,*c*taking all at a time are:

(*abc*,*acb*,*bac*,*bca*,*cab*,*cba*)

- Number of Permutations:Number of all permutations of
*n*things, taken*r*at a time, is given by:^{n}P_{r}=*n*(*n*- 1)(*n*- 2) ... (*n*-*r*+ 1) =*n*!( *n*-*r*)!Examples:^{6}P_{2}= (6 x 5) = 30.^{7}P_{3}= (7 x 6 x 5) = 210.- Cor. number of all permutations of
*n*things, taken all at a time =*n*!.

- An Important Result:If there are
*n*subjects of which*p*_{1}are alike of one kind;*p*_{2}are alike of another kind;*p*_{3}are alike of third kind and so on and*p*_{r}are alike of*r*^{th}kind,

such that (*p*_{1}+*p*_{2}+ ...*p*_{r}) =*n*.Then, number of permutations of these *n*objects is =*n*!( *p*_{1}!).(*p*_{2})!.....(*p*_{r}!) - Combinations:Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.Examples:
- Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
- All the combinations formed by
*a*,*b*,*c*taking*ab*,*bc*,*ca*. - The only combination that can be formed of three letters
*a*,*b*,*c*taken all at a time is*abc*. - Various groups of 2 out of four persons A, B, C, D are:AB, AC, AD, BC, BD, CD.
- Note that
*ab**ba*are two different permutations but they represent the same combination.

- Number of Combinations:The number of all combinations of
*n*things, taken*r*at a time is:^{n}C_{r}=*n*!= *n*(*n*- 1)(*n*- 2) ... to*r*factors. ( *r*!)(*n*-*r*)!*r*!Note:^{n}C_{n}= 1 and^{n}C_{0}= 1.^{n}C_{r}=^{n}C_{(n - r)}

Examples:i. ^{11}C_{4}=(11 x 10 x 9 x 8) = 330. (4 x 3 x 2 x 1) ii. ^{16}C_{13}=^{16}C_{(16 - 13)}=^{16}C_{3}=16 x 15 x 14 = 16 x 15 x 14 = 560. 3! 3 x 2 x 1

# Permutation and Combination - Important Formulas |Aptitude

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