## Permutation and Combination Formulas

**1. Factorial Notation:**

Let

*n*be a positive integer. Then, factorial*n*, denoted*n*! is defined as:*n! = n(n – 1)(n – 2) … 3.2.1.*

*Examples:*- We define
*0! = 1*. - 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.

**2. Permutations:**

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

*Examples:*

- All permutations (or arrangements) made with the letters
*a*,*b*,*c*by taking two at a time are ().*ab*,*ba*,*ac*,*ca*,*bc*,*cb* - All permutations made with the letters
*a*,*b*,*c*taking all at a time are:

(*abc*,*acb*,*bac*,*bca*,*cab*,*cba*

**3. Number of Permutations:**

Number of all permutations of

*n*things, taken*r*at a time, is given by:

*Examples:*^{6}P_{2}= (6 x 5) = 30.^{7}P_{3}= (7 x 6 x 5) = 210.*Cor. number of all permutations of**n*things, taken all at a time =*n*!.

**4. An Important Result:**

If there are

such that (

*n*subjects of which*p*_{1}are alike of one kind;*p*_{2}are alike of another kind;*p*_{3}are alike of third kind and so on and*p*_{r}are alike of*r*^{th}kind,such that (

*p*_{1}+*p*_{2}+ …*p*_{r}) =*n*.
Then, number of permutations of these

*n*objects is =**5. Combinations:**

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a

*combination*.

*Examples:*- Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
- All the combinations formed by
*a*,*b*,*c*taking.*ab*,*bc*,*ca* - The only combination that can be formed of three letters
*a*,*b*,*c*taken all at a time is.*abc* - Various groups of 2 out of four persons A, B, C, D are:
*AB, AC, AD, BC, BD, CD*. - Note that
*ab**ba*are two different permutations but they represent the same combination.

**6. Number of Combinations:**

The number of all combinations of

*n*things, taken*r*at a time is:

*Note:*^{n}C_{n}= 1 and^{n}C_{0}= 1.^{n}C_{r}=^{n}C_{(n – r)}

*Examples:*
i.

^{11}C_{4}=
ii.

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