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

3. $n! + 1$ is not divisible by any natural number between $2 \text{ and } n$

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

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 $\displaystyle p+q = n \text{ is } \frac{n!}{p! q!}$

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 $\displaystyle p_1+p_2+p_3+\ldots +p_r = n \text{ is } \frac{n!}{p_1! p_2! p_3! \ldots p_r!}$

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 $\displaystyle \frac{n!}{p! q!}$

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 $\displaystyle \frac{(m + n)!}{m ! n !}$ .

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

(ii) The number of ways of dividing $2n$ different elements into two groups of $n$ objects each is $\displaystyle \frac{(2n)!}{(n!)^2}$ , 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 $\displaystyle \frac{(3n)!}{((n!)^3}$ .

(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 $\displaystyle \frac{(2n)!}{2!(n!)^2}$.

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

(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 $\displaystyle \frac{(mn)!}{(n!)^m m!}$

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

Circular Permutation

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

Case 1: – Clockwise and Anticlockwise orders are different. $\text{Formula: } P_n = (n-1)!$

Case 2: – Clockwise and Anticlockwise orders are the same. $\displaystyle \text{Formula: } P_n = \frac{(n-1)!}{2!}$