Table of Contents

Toggle## Factorial

For any natural number n, we define factorial as

n! = n(n-1)(n-2)…3×2×1

**Important Results Related to Factorial**

- 0! = 1! = 1
- Factorials of negative integers and fractions are not defined.
- n! = n(n-1)! = n(n-1)(n-2)!

## Permutations

The number of ways of arranging n distinct objects taking some or all of a number of things is called a permutation.

P(n, r) = ^{n}P_{r} = n!\over (n-r)!

### Properties of permutation

^{n}P_{n}= n(n-1)(n-2) …. 3×2×1^{n}P_{o}= n!\over n!^{n}P_{o}= n^{n}P_{n – 1}= n!^{n – 1 }P_{r}+ r.^{n – 1 }P_{r – 1}=^{n}P_{r}- {{^nP_r}\over{^nP_{r-1}}}=n-r+1

### Important Results on Permutation

- The number of permutations of n different things taken r at a time, when each thing may be repeated any number of times is n
^{r}. - The number of permutations of n things taken all at a time, in which p are alike of one kind, q are alike of the second kind and r are alike of the third kind and rest are different is

n!\over p!q!r!

### Restricted Permutations

(i) Number of permutations of n different things taken r at a time,

- When a particular thing is to be included in each arrangement is r.
^{n-1}P_{r-1}. - When a particular thing is always excluded is
^{n-1}P_{r}.

(ii) Number of permutations of n different objects taken r at a time in which m particular object are always

- excluded =
^{n-m}P_{r} - included =
^{n-m}P_{r-m}× r!

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

### Circular Permutations

A circular permutation is an arrangement of objects in a circle, where the order of the objects is important but the starting position is not.

- The number of circular permutations of n different things taken all at a time is (n-1)!. If clockwise and anticlockwise orders are taken as different.
- The number of circular permutations of n different things taken all at a time when clockwise and anticlockwise orders are not taken as different = {1\over2} (n-1)!

## Combination

A combination is a selection of items from a set where the order of the selection does not matter.

C (n, r) or ^{n}C_{r} = n!\over r!(n - r)! 0≤ r ≤ n

### Properties of combination

^{n}C_{o}=^{n}C_{n}= 1^{n}C_{1}= n^{n}C_{r}=^{n}C_{n-r}- If
^{n}C_{a}=^{n}C_{b}, then either a = b or a + b = n ^{n}C_{r}= ^nP_r\over r!^{n}C_{r}+^{n}C_{r – 1}=^{n+1}C_{r}- { {^n}C_r \over{^{n-1}}C_{r-1}}={n\over r}
- { {^n}C_r \over{^{n}}C_{r-1}}={n-r+1\over r}
^{n}C_{0}+^{n}C_{1}+^{n}C_{2}+^{n}C_{3}+ … +^{n}C_{n}= 2^{n}^{n}C_{0}+^{n}C_{2}+^{n}C_{4}+ … =^{n}C_{1}+^{n}C_{3}+^{n}C_{5}+ … = 2^{n-1}

### The important result on the combination

- The number of combinations of n different things have taken r at a time allowing repetitions is
^{n+r-1}C_{r}. - The total number of combinations of n different objects taken r at a time in which

(a) m particular objects are excluded =^{n-m}C_{r}

(b) m particular objects are included =^{n-m}C_{r-m}

## Selection from different Items

- The number r of ways of selecting at least one item from n distinct items is 2
^{n-1}. - The number of ways of answering one or more of n questions is 2
^{n }– 1. - The number of ways of answering one or more of n questions when each question has an alternative = 3
^{n-1}.

## Division into groups

- The number of ways in which (n+m) different things can be divided into two groups which m and n things respectively = (m+n)!\over m!n!, where m ≠ n
- The number of ways of dividing 2n different elements into two groups of n objects each is (2n)!\over (n!)^2, when the distinction can be made between the groups.
- The number of ways of dividing 2n different elements into two groups of n objects when no distinction can be made between the groups = (2n)!\over 2(n!)^2