Quants Permutation & Combination

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:

1. We define 0! = 1.
2. 4! = (4 x 3 x 2 x 1) = 24.

• 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:

1. All permutations (or arrangements) made with the letters a, b, cby taking two at a time are (ab, ba, ac, ca, bc, cb).
2. All permutations made with the letters a, b, ctaking 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:

Examples:

30. ⁶P₂= (6 x 5) = 30.
31. ⁷P₃= (7 x 6 x 5) = 210.

• number of all permutations of nthings, 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.

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, ctaking ab, bc, ca.

3. The only combination that can be formed of three letters a, b, ctaken 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 abba 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:

• . ⁿC_n= 1 and ⁿC₀ = 1.

1. ⁿC_r= ⁿC_{(n – r)}

Examples: