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.
3! = (3 x 2 x 1) = 6.

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 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:
ⁿP_r = n(n - 1)(n - 2) ... (n - r + 1) = n!/(n - r)!
Examples:
⁶6P₂ = (6 x 5) = 30.
Cor. number of all permutations of n things, taken all at a time = n!.

4. An Important Result:
If there are n subjects of which p₁ are alike of one kind; p₂ are alike of another kind; p₃ are alike of third kind and so on and p_r are alike of r^th kind,
such that (p₁ + p₂ + ... p_r) = n.
Then, number of permutations of these n objects is = n!/(p₁!).(p₂!).....(p_r!)

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:
i. 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.
ii. All the combinations formed by a, b, c taking 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.

6. Number of Combinations:
The number of all combinations of n things, taken r at a time is:
ⁿC_r = n!/(r!)(n - r)! = n(n - 1)(n - 2) ... to r factors/r! .

Note:
ⁿC_n = 1 and ⁿC₀ = 1.
ⁿC_r = ⁿC_{(n - r)}
Examples:
i. ¹¹C₄ = (11 x 10 x 9 x 8)/(4 x 3 x 2 x 1) = 330.

ii. ¹⁶C₁₃ = ¹⁶C_{(16 - 13)} = ¹⁶C₃ = 16 x 15 x 14/3! = 16 x 15 x 14/3 x 2 x 1 = 560.