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

** n! = n(n-1)(n-2) ........ 3.2 .1**

**Examples:- 5! = (5 x 4 x 3 x 2 x 1) = 120; 4! = (4 x 3 x 2 x 1) = 24 etc.,**

**We Define, 0! = 1 **

**Permutations:- **The different arrangements of a given aumber of things by taking some or all at a time, are called **permutations**.

**Ex:- 1** All permutations (or arrangements) made with the letters a, b, c by taking two at a time are ** (ab, ba, ac, ca, bc, cb).**

**Ex:- 2 **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 things, taken r at a time, is given by

\(^{n} \mathrm{P}_{r}\) = \(n(n-1)(n-2) \dots \dots(n-r+1)\)= \(\frac{n !}{(n-r) !}\)

**Examples:- **\(^{6} \mathrm{P}_{2}\) = (6 x 5) = 30; \(^7P_3\) = ( 7 x 6 x 5) = 210.

**Cor. Number of all permutations of things, taken all at a time = n!**

**An Important Result:- **If there are n objects of which \(p_{1}\) are alike of one kind; \(p_{2}\) are alike if another kind; \(p_{3}\) are alike of the third kind and so on and \(p_r\) are alike of rth kind, such that \(\left(p_{1}+p_{2}+\dots \ldots+p_{r}\right)\) = n

**Then ****the number**** of permutations of these n objects is \(\frac{n !}{\left(p_{1} !\right) \cdot\left(p_{2} !\right) \ldots \dots\left(p_{r} !\right)}\)**

**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.

**Ex:- 1 **Suppose we want to select two out of three boys A, B, C Then, possible selections are **AB, BC, and CA**

Note that AB and BA represent the same selection.

**Ex:- 2 **All the combinations formed by a, b, c, taking two at a time are **ab, bc, ca.**

**Ex:- 3 **The only combination that can be formed of three letters a, b, c taken all at a time is **abc.**