Permutations with example problem

Introduction:

         The process of selecting things is called as combination and that of arranging things is called as permutation.

A few problems related to the permutations are,

1. Arrangement of books on a shelf.
2. Formation of numbers with the given digits;
3. Formation of words with the given letters.

To find the number of permutations of n object taken r at a time denoted by ⁿP_r. The letter P is an abbreviation of the word ‘permutations’.

Permutations Example Problem

Example 1:

Suppose the six digits 1, 2, 4, 5, 6, 7 are given to us and we have to find the total number of number which can be formed under different condition.

Condition 1.      There is no restriction:

The number of 6-digits numbers = ⁶P₆ = 6! = 6x5x4x3x2x1=7200

Condition 2.      The numbers in which a particular digit occupies a particular place.

Suppose we have to form a number in which 5 always occur in the ten’s place. In this case the ten’s place is fixed and reaming five place can be filled in by the reaming 5 digit in ⁵P_5, i.e. 5! = 5x4x3x2x1 = 120.

Condition 3.      Number divisible by a particular number

Suppose we have to form a numbers which may be divisible by 2. These numbers will have 2 or 4 or 6 in unit place. Thus the unit’s place can be filled in 3 ways. After having filled up the unit’s place in any one of the above ways, the remaining five place can be filled in ⁵P₅ = 5! = 120.

The total number of numbers divisible by 2 = 120 x 3 = 360.