Permutation And Combination

Introduction

Factorial Notation:-  Let n be a positive integer. Then, factorial n, denoted by n! is defined as 

              n! = n(n-1)(n-2) ........ 3.2 .1

Examples:- 5! = (5 x 4 x 3 x 2 x 1) = 120; 4! = (4 x 3 x 2 x 1) = 24 etc.,

We Define, 0! = 1 

Permutations:- The different arrangements of a given aumber of things by taking some or all at a time, are called permutations.

Ex:- 1 All permutations (or arrangements) made with the letters a, b, c by taking two at a time are  (ab, ba, ac, ca, bc, cb).

Ex:- 2 All permutations made with the letters a, b, c, taking all at a time are (abc, acb, bac, bca, cab, cba)

Number of Permutations and Important Results

Number of Permutations:-  Number of all permutations of n things, taken r at a time, is given by 

                   \(^{n} \mathrm{P}_{r}\) = \(n(n-1)(n-2) \dots \dots(n-r+1)\)\(\frac{n !}{(n-r) !}\)

Examples:- \(^{6} \mathrm{P}_{2}\) = (6 x 5)  = 30; \(^7P_3\) = ( 7 x 6 x 5) = 210.

Cor. Number of all permutations of things, taken all at a time = n!

An Important Result:- If there are n objects of which \(p_{1}\) are alike of one kind; \(p_{2}\) are alike if another kind; \(p_{3}\) are alike of the third kind and so on and \(p_r\) are alike of rth kind, such that \(\left(p_{1}+p_{2}+\dots \ldots+p_{r}\right)\) = n

Then the number of permutations of these n objects is \(\frac{n !}{\left(p_{1} !\right) \cdot\left(p_{2} !\right) \ldots \dots\left(p_{r} !\right)}\)

Combinations

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.

Ex:- 1 Suppose we want to select two out of three boys A, B, C Then, possible selections are AB, BC, and CA

Note that AB and BA represent the same selection.

Ex:- 2 All the combinations formed by a, b, c, taking two at a time are ab, bc, ca.

Ex:- 3 The only combination that can be formed of three letters a, b, c taken all at a time is abc.