Aptitude ➤ Permutations And Combinations ➤ Set 1

#### Permutations And Combinations : Important Facts and 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.

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:
i. 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.
ii. All the combinations formed by a, b, c taking ab, bc, ca.
iii. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
iv. Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
v. 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! = n(n - 1)(n - 2) ... to r factors . (r!)(n - r)! r!

Note:

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

Examples:

 i. 11C4 = (11 x 10 x 9 x 8) = 330. (4 x 3 x 2 x 1)
 ii. 16C13 = 16C(16-13) = 16C3 = 16 x 15 x 14 = 16 x 15 x 14 = 560. 3! 3 x 2 x 1
Question 6
Q6.  How many words can be formed from the letters of the word 'EXTRA' so that the vowels are never together?
Question 7
Q7.  In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?
Question 8
Q8.  How many words can be formed from the letters of the word 'DIRECTOR' so that the vowels are always together?
Question 9
Q9.  In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?
Question 10
Q10.  In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women? 