Basic Permutations - Permutations and Combinations

Permutations

Each of the different arrangement which can be made by taking some or all of a number of things at a time is called a permutation.

For example: Let's say we have three letters: A, B, and C. we want to find out how many different ways we can arrange these letters by taking some or all of them at a time.

So, by taking some or all of the letters A, B, and C at a time, we can form the following permutations:

• Single-letter permutations: A, B, C (3 permutations)
• Two-letter permutations: AB, AC, BA, BC, CA, CB (6 permutations)
• Three-letter permutations: ABC, ACB, BAC, BCA, CAB, CBA (6 permutations)

In total, there are 15 different permutations that can be made using the letters A, B, and C by taking some or all of them at a time.

Notation: The number of permutaions of n thing taken r at a time is denoted by ⁿP_r or P(n,r). This letter P is an abbreviation of the word permutation

Basic permutations formula:-

• ⁿP_n or P(n,n) = n!
• ⁿP_r or P(n,r) = $\frac{\text{n!}}{\text{(n-r)!}}$

Here, P stands for permutations, n stands for total number of things and r stands for things taken at the time.

Practice Question

Question 1. Amit wants to arrange 3 Economics, 2 History and 4 English books on a shelf. If the books on same subject are different, determine the number of possible arrangement.

Solution:

Total number of Economics books = 3
Total number of History books = 2
Total number of English books = 4
Since the books of each subject are also different we can consider each book as 1 irrespective of its subject so total number of books = 9.
total number of arrangements = ⁹P₉ = $\frac{\text{9!}}{\text{(9-9)!}}$
= 9!

Question 2. From a pool of 12 candidates, in how many ways can we select president, vice president, secretary and a treasurer if each of the 12 candidate can hold any office?

Solution:

Total number of candidates,n = 12.

Total number of posts, r = 4.

Total number of ways 12 candidates can be selected in these four posts = ¹²P₄

$\text{=}\frac{12!}{(12-4)!}$

$\text{=}\frac{12!}{(8!}$

$\text{=}\frac{12*11*10*9*8!}{8!}$

=11880