# 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.

Example

- We define 0! = 1.
- 4! = (4 Ã— 3 Ã— 2 Ã— 1) = 24.
- 5! = (5 Ã— 4 Ã— 3 Ã— 2 Ã— 1) = 120.

# Permutations

Example

- 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)

# Number of Permutations

Example

^{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!.

# An Important Result

_{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

# 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.

Example

- 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:**

^{n}C_{n}= 1 and^{n}C_{0}= 1.^{n}C_{r}=^{n}C_{(n - r)}

Example

^{16}C_{13}=^{16}C_{(16 - 13)}=^{16}C_{3}=