1. Permutations & Combinations
A permutation is an act of arranging numbers or objects in order.
For example, if we arrange the letters C, A and P, we get 6 different permutations as follows:
CAP ACP PAC
CPA APC PCA
Note that the position of letters is important in a permutation.
The way of selecting the objects from a group of objects is called combinations.
For example, if we wish to choose 2 letters from the group C, A and P, we get 3 combinations as follows:
CA CP AP
Here AC = CA because the position of these letters is not important in a combination.
1.1 How to Calculate Permutations & Combinations?
Listing method:
Example: How many different four-digit numbers can be made by using the digits 2, 4, 6 and 8 assuming each digit can be used only once?
Solution:
List down all the three-digit numbers possible using the digits 2, 4 and 8
248 284 482
284 842 824
6 different three-digit numbers can be made. This is the case of permutation as the position of digits is important here.
Example: How many groups can be made from the word "MATHS" if each group consists of 3 alphabets?
Solution:
List down all the possible groups that can be made using the word ‘MATHS’
MAT MAH MAS HAT
AHS THS TSM HSM
HMT SAT
Total number of groups = 10
Here MAT = ATM = TAM = TMA as the position is not important here. This is a simple example of a combination.
1.2 Seating Arrangement: Solved Example
There are three people A, B and C and two chairs. In how many ways these three people can occupy the two chairs?
Solution:
Let the boxes below represent two chairs.
List down all the possibilities.
The total number of possibilities = 6
This is the case of permutation as the position of people (A, B and C) is important here.
2. Formula to find Permutations & Combinations
n! is the number of ways of arranging n unlike objects in a line.
n!(pronounced 'n factorial') = n × (n-1) × (n-2) × .... 3 × 2 × 1
Permutation formula:
The number of ways of arranging n objects, of which a of one type are alike, b of second type is alike, etc is:
n! ÷ a! b! .....
The number of ways of arranging n unlike objects in a circle or in a ring when clockwise and anticlockwise arrangements are different is (n-1)! When anti-clockwise and clockwise arrangements are the same, the number of ways is 1⁄2(n-1)!
Let us now learn how to apply these formulas using a few examples.
2.1 Solved Examples using formulas
Example: How many six-digit numbers can be formed using the digits given below:
3, 5,6,6,7,8
Solution: Number of digits to be arranged i.e. n = 6
Number of times digit 3 is repeating i.e. a = 2
Using permutation formula,
Number of ways in which 6 digits can be arranged
= n! ÷ a!
= 6! ÷ 2! = 6 × 5 × 4 × 3 × 2 × 1 ÷ 2 × 1 = 6 × 5 × 4 × 3 = 360
360 six-digit numbers can be formed using the given digits.
Example: Work out the number of ways in which 6 women can be arranged at the round table.
Solution: Number of women to be arranged at the round table i.e. n = 6
Using permutation formula for arranging objects in a circle,
Number of ways = (n - 1)!
= (6 - 1)!
= 5!
= 5 × 4 × 3 × 2 × 1
=120
6 women can be arranged at the round table in 120 ways.
