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:

 i. We define 0! = 1. ii. 4! = (4 x 3 x 2 x 1) = 24. iii. 5! = (5 x 4 x 3 x 2 x 1) = 120.
2.  Permutations:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Examples:

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

nPr = n(n - 1)(n - 2) ... (n - r + 1) =
 n! (n - r)!

Examples:

 i. 6P2 = (6 x 5) = 30. ii. 7P3 = (7 x 6 x 5) = 210. iii. 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 pr are alike of rth kind,
such that (p1 + p2 + ... pr) = n.

Then, number of permutations of these n objects is =
 n! (p1!).(p2)!.....(pr!)

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:

1.    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.
2.    All the combinations formed by a, b, c taking ab, bc, ca.
3.    The only combination that can be formed of three letters a, b, c taken all at
a time is abc.

4.    Various groups of 2 out of four persons A, B, C, D are:

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

5.    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:

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

Note:

 i. nCn = 1 and nC0 = 1. ii. nCr = 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) = 16C13 =
 16 x 15 x 14 3!
=
 16 x 15 x 14 3 x 2 x 1
= 560.