Permutation and Combination Formula

Permutation and Combination Formula

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.

POINTS TO REMEMBER

0! = 1.

1! = 1.

2! = 2.

3! = 6.
4! = 24.
5! = 120.

6! = 720.

7! = 5040.

8! = 40320.

9! = 362880.

Permutations:

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

Examples:

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

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:

6P2 = (6 x 5) = 30.

7P3 = (7 x 6 x 5) = 210.

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 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!)

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:

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: