Permutation: Each of the different arrangement which can be made by taking some or all of a number of things is called a permutation.

1. The continued product of first natural numbers is called and is denoted by .

Therefore

Factorials of proper fractions and negative integers are not defined.

2.

3. is not divisible by any natural number between and

4. If is a natural number and r is a positive integer such that , then

5.

i) Let and be positive integers such that , then the number of permutations of distinct objects taken at a time is

ii) The number of all permutations ( arrangements) of distinct objects taken all at a time is .

iii) The number of mutually distinguishable permutations of things taken all at a time, of which are alike of one kind and are alike of second kind such that is

iv) The number of mutually distinguishable permutations of things taken all at a time, of which are alike, are alike, are alike, are alike, such that is

v) The number of mutually distinguishable permutations of things of which are alike of one kind and are alike of second kind and rest are distinct is

vi) Number of permutations of n different things taken all at a time, when m specified things always come together is .

vii) Number of permutations of n different things taken all at a time, when m specified things never come together is .

Fundamental Principles of Counting

Multiplication Principle: If first operation can be performed in ways and then a second operation can be performed in ways. Then, the two operations taken together can be performed in ways. This can be extended to any finite number of operations.

Addition Principle: If first operation can be performed in ways and another operation, which is independent of the first, can be performed in ways. Then, either of the two operations can be performed in ways. This can be extended to any finite number of exclusive events.

Division into Groups

(i) The number of ways in which different things can be divided into two groups which contain and things respectively .

This can be extended to different things divided into three groups of things respectively .

(ii) The number of ways of dividing different elements into two groups of objects each is , when the distinction can be made between the groups, i.e., if the order of group is important. This can be extended to different elements into groups is .

(iii) The number of ways of dividing different elements into two groups of object when no distinction can be made between the groups i.e., order of the group is not important is .

This can be extended to different elements into groups is .

(iv) The number of ways in which different things can be divided equally it into groups, if order of the group is not important is

(v) If the order of the group is important, then number of ways of dividing different things equally into distinct groups is .

Circular Permutation

Circular permutation is the total number of ways in which distinct objects can be arranged around a fix circle. It is of two types.

**Case 1: **– Clockwise and Anticlockwise orders are different.

Formula:

**Case 2: **– Clockwise and Anticlockwise orders are same.

Formula: