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:

1. We define 0! = 1.

2. 4! = (4 x 3 x 2 x 1) = 24.

3. 5! = (5 x 4 x 3 x 2 x 1) = 120.

Permutations:

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

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:

Examples:

1. ⁶P₂ = (6 x 5) = 30.

2. ⁷P₃ = (7 x 6 x 5) = 210.

3. Cor. number of all permutations of n things, taken all at a time = n!.

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.

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.

Number of Combinations:

The number of all combinations of n things, taken r at a time is:

Note:

1. ⁿC_n = 1 and ⁿC₀ = 1.

2. ⁿC_r = ⁿC_{(n - r)}

Examples: