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 n natural numbers is called \text{"n factorial} and is denoted by n! .

Therefore n! = 1 \times 2 \times 3 \times 4 \times 5 \times \ldots \times n

Factorials of proper fractions and negative integers are not defined.

0! = 1

2. \frac{(2n)!}{n!} = 1 \cdot 3 \cdot 5 \ldots (2n-1) 2^n

3. n! + 1 is not divisible by any natural number between 2 and n

4. If n is a natural number and r is a positive integer such that 0 \leq r \leq n , then ^{n} \rm P_{r} = \frac{n!}{(n-r)!}

5.

i) Let r and n be positive integers such that 1 \leq r \leq n , then the number of permutations of n distinct objects taken r at a time is n(n-1)(n-2) \ldots (n-(n-r))

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

iii) The number of mutually distinguishable permutations of n things taken all at a time, of which p are alike of one kind and q are alike of second kind such that p+q = n is \frac{n!}{p! q!}

iv) The number of mutually distinguishable permutations of n things taken all at a time, of which p_1 are alike, p_2 are alike, p_3 are alike, \ldots p_r are alike,  such that p_1+p_2+p_3+\ldots +p_r = n is \frac{n!}{p_1! p_2! p_3! \ldots p_r!}

v) The number of mutually distinguishable permutations of n things of which p are alike of one kind and q are alike of second kind and rest are distinct is \frac{n!}{p! q!}

vi) Number of permutations of n different things taken all at a time, when m specified things always come together is m!(n - m + 1)! .

vii) Number of permutations of n different things taken all at a time, when m specified things never come together is n! - m! \times  (n - m + 1)! .

Fundamental Principles of Counting

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

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

Division into Groups

(i) The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively \frac{(m + n)!}{m ! n !} .

This can be extended to (m + n + p) different things divided into three groups of m, n, p things respectively \frac{(m + n + p)!}{m!n! p!} .

(ii) The number of ways of dividing 2n different elements into two groups of n objects each is \frac{(2n)!}{(n!)^2} , when the distinction can be made between the groups, i.e., if the order of group is important. This can be extended to 3n different elements into 3 groups is \frac{(3n)!}{((n!)^3} .

(iii) The number of ways of dividing 2n different elements into two groups of n object when no distinction can be made between the groups i.e., order of the group is not important is \frac{(2n)!}{2!(n!)^2} .

This can be extended to 3n different elements into 3 groups is \frac{(3n)!}{3!(n!)^3} .

(iv) The number of ways in which mn different things can be divided equally it into m groups, if order of the group is not important is \frac{(mn)!}{(n!)^m m!}

(v) If the order of the group is important, then number of ways of dividing mn different things equally into m distinct groups is \frac{(mn)!}{(n!)^m} .

Circular Permutation

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

Case 1: – Clockwise and Anticlockwise orders are different.

Formula: P_n = (n-1)!

Case 2: – Clockwise and Anticlockwise orders are same.

Formula: P_n = \frac{(n-1)!}{2!}