**Fractional Notation**

Let n be a positive integer. Then, factorial n is denoted by n! is defined as:

n! = n(n - 1)(n - 2)(n - 3).....3.2.2.

**Examples:**

(i) 6! = (6 x 5 x 4 x 3 x 2 x 1) = 720

(ii) 5! = (5 x 4 x 3 x 2 x 1) = 120

(ii) 4! = (4 x 3 x 2 x 1) = 24

we define, 0! = 1.**Permutations**

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

**Examples:**

All permutations made by with 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.****Number of Permutations**

Number of all permutations of n objects is:

^{n}P_{r}= n(n - 1)(n - 2)..... (n- r -1) =n!

(n - r)!

**Examples:**

(i)^{9}P_{3}= (9 x 3) = 27.

(ii)^{8}P_{5}= (8 x 5) = 40.

**Note:**Cor. Number of all permutations of n things, taken all at a time = n!.**An Important Result**

If there are 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 rth kind,

such that (p_{1}+_{2}+ ....._{r}) = n.

Then, number of permutations of these n objects is = n!

(p_{1}!)(p_{2}!) ... (p_{r}!)

**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:**

(i) Suppose there are three boys A,B, C, and we want to select two out of three, then the combinations are:**AB, BC, and CA**.

Note: AB and BA represent the same selection.

(ii) All the combinations formed by a, b, c, taking two at a time are**ab, bc, ca**. (iii) The only combination that can be formed of three letters a, b, c taken all at a time is**abc**. (iv) Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

(v)Note that ab ba are two different permutations but they represent the same combination.**Number of Combinations**

The number of all combinations of n things, taken r at a time is:

^{n}C_{r}=n!

r!(n - r)!= n(n - 1)(n - 2).....to r factors

r!

**Note:**

^{n}C_{n}= 1 and^{n}C_{0}= 1.

**An Important Result:**^{n}C_{r}=^{n}C_{(n - r)}

**Examples:**

(i)^{11}C_{4}=11 x 10 x 9 x 8

4 x 3 x 2 x 1= 330.