**Formula Sheet On Permutation and combination**

** Fundamental Principle of Counting:**

1. Rule of Sum:- A task is performed in m ways and another task is performed in n ways and both tasks cannot be performed simultaneously. So, either task can be accomplished in (m + n) ways.

2. Rule of Multiplication:- There are two tasks, A and B can be performed in m and n ways respectively. So, the number of different ways of doing both tasks A and B simultaneously is (m × n) ways.

**Factorial:** Factorial is a notation for multiplication of consecutive integers. The factorial is represented by the symbol ‘!’

n! = 1 × 2 × 3 × 4 × …. × (n – 1) × n (n! mean multiplication of first n natural numbers)

Ex: 4! = 1 x 2 x 3 x 4

n! = n × (n – 1)!

Ex: 4! = 4 x (4 – 1)! = 4 x 3!

Value of the few frequently used factorial.

0! = 1

1! = 1

2! = 2 x 1 = 2

3! = 3 x 2 x 1 = 6

4! = 4 x 3 x 2 x 1 = 24

5! = 5 x 4 x 3 x 2 x 1 = 120

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 and so on.

**Permutation:** Permutation means arranging i.e. “selecting and ordering” one or more objects from the given certain objects (maybe alike or different). The number of permutations of n different objects taken r at a time is represented as :

n(n – 1)(n – 2) ……… (n – r + 1) (where, 0 ≤ r ≤ n)n(n – 1)(n – 2) ……… (n – r + 1) (where, 0 ≤ r ≤ n)In Permutation and Combination, Combination is selection and Permutation is selection as well as arrangements.

**Combination:** Combination means selecting one or more objects from the given certain objects (maybe alike or different). The combination of n distinct objects taken r at a time is represented and calculated as:

Here, r can be any positive integer less than or equal to n.

Ex: In a bag, there are 3 red, 4 black and 6 green balls. In how many different ways, we can select 2 red balls?

Sol: Required number of ways = 3C2

Ex: In a bag, there are 3 red, 4 black and 6 green balls. In how many different ways, we can select 2 balls such that none of them is red?

Sol: Required number of ways = (4 + 6)C2 = 10C2

Ex: In a bag, there are 3 red, 4 black and 6 green balls. In how many different ways, we can select 2 balls such that both are of the same colour?

Sol: Required number of ways = 3C2 + 4C2 + 6C2

Ex: In a bag, there are 3 red, 4 black and 6 green balls. In how many different ways, we can select 2 balls such that both are of different colours?

Sol: To select 2 balls of different colours, we can select in the combination of (red, black), (black, green), (green, red)

Required number of ways = (3 × 4) + (4 × 6) + (6 × 3) = 54

Ex: In how many different ways can the letters of the word 'ROSTED' be arranged?

Sol: Required number of ways = 6!

**Permutation of Alike Objects:** The number of permutations of n objects taken all at a time in which, p are alike objects of one kind, q are alike objects of second kind & r are alike objects of a third kind and the rest

Ex: In how many different ways can the letters of the word 'MOTION' be arranged?

Sol: Here, the total number of letters = 6 O – 2 (alike) and M, T, I, N

Ex: In how many different ways can the letters of the word 'FRUSTRATION' be arranged?

Sol: Here, the total number of letters = 11

R – 2, T – 2 and F, U, S, A, I, O, N

Ex: In How many different words can be formed with the letters of the word 'POLICE' beginning with P?

Sol: In this case, we will arrange the other 5 digits (except P, because P is fixed) Hence, the required number of ways = 5!

Ex: In How many different words can be formed with the letters of the word 'POLICE' beginning with P and ending with E?

Sol: In this case, we will arrange the other 4 digits (except P and E as both are fixed) Hence, the required number of ways = 4!

**To get access to download complete formula sheets on Probability, click on the link below -**

**Download PDF for Formula Sheets: ****Permutation and combination**

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