**Factorial Notation**

Let

*n**be a positive integer. Factorial of**n**denoted by**n***!**and is defined as**n! = n(n - 1)(n - 2)(n-3) ... 3.2.1.**

*Examples:*

**We define 0! = 1.**

**2! = 2.1 = 2**

**3! = 3.2.1 = 6**

**4! = 4.3.2.1 = 24**

**5! = 5.4.3.2.1 = 120**

*So on......*

**Permutations (**

*Arrangements*)
The different arrangements
of a given number or things by taking some or all at a time, are called
permutation.

*Examples:*

**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).

**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}**P**

_{r}=*n*(*n*- 1)(*n*- 2) ... (*n*-*r*+ 1)*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!.*

*Note:*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

*r*

^{th}kind,

such that (

*p*

_{1}+

*p*

_{2}+ ...

*p*

_{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:*

- 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.*

**Number of Combinations:**

The
number of all combinations of

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

**1.**

^{n}**C**

_{n}= 1 and^{n}C_{0}= 1.**2.**

^{n}**C**

_{r}=^{n}C_{(n - r)}*Examples:*

^{11}C_{4} = |
(11 x 10 x 9 x 8) |
= 330. |

(4 x 3 x 2 x 1) |

^{16}C_{13} = ^{16}C_{(16
- 13)} = ^{16}C_{3} = |
16 x 15 x 14 |
= |
16 x 15 x 14 |
= 560. |

3! |
3 x 2 x 1 |

Choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations.

16! | = | 16! | = | 16! | = 560 |

3!(16-3)! | 13!(16-13)! | 3!×13! |