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Permutation and Combination

Important Formulae

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.

2. 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!

3. 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!

General Tips

1.Factorial Notation:

We define 0! = 1.
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:

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

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

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.

6.Number of Combinations:
Note:

nCn = 1 and nC0 = 1.
nCr = nC(n - r)

Example

1. 11C4 = (11*10*9*8)/(1*2*3*4) = 330
2. 16C13 = 16C(16-13) = 16C3 =(16*15*14)/(1*2*3) = 560