Introduction to Basics of Permutation and Combination

In today’s blog, we explore the fundamental concepts of Permutation and Combination (P & C) — an essential part of Engineering Mathematics after understanding the basics of counting.

What is Permutation and Combination?

Permutation is the arrangement of items in a particular order.

Combination is the selection of items, where the order does not matter.

Simple Trick to Remember:

Whenever you only have to select objects, use Combination.
Whenever you have to select and arrange, use Permutation.

Important: If repetition is allowed, use the Multiplication Principle instead of standard permutation formulas.

Formulae You Must Know

Permutation (nPr):

{}^nP_r = \frac{n!}{(n-r)!}

Combination (nCr):

{}^nC_r = \frac{n!}{r!(n-r)!}

Where:

n = total number of items
r = number of items selected

Where to Use Combination?

Keywords that indicate Combination:

Formation of a team / group / committee
Handshakes between people
Selection of students or employees
Choosing books or objects

Where to Use Permutation?

Keywords that indicate Permutation:

Formation of words from letters
Formation of numbers
Seating arrangements
Arranging people in a line

Practice Questions with Answers

Question 1:

How many handshakes are possible between 5 people?

{}^5C_2 = 10

Answer: 10 handshakes

Question 2:

If there are 66 handshakes in a party, how many people were there?

{}^nC_2 = 66 \implies n(n-1) = 132 \implies n = 12

Answer: 12 people

Question 3:

How many 4-digit numbers can be formed using digits 1–9?

(a) Without repetition

{}^9P_4 = 9 \times 8 \times 7 \times 6 = 3024

(b) With repetition allowed

\[ 9^4 = 6561 \]

Answer: (a) 3024 (b) 6561

Question 4:

A committee of 6 is to be selected from 5 boys and 6 girls.

(a) No restriction

{}^{11}C_6 = 462

(b) At least 2 girls → 456 ways

Question 5:

In how many ways can 5 different books be arranged on 3 shelves?

{}^5P_3 = 60

Answer: 60 ways

Conclusion

Understanding when to use Combination (selection only) vs Permutation (selection + arrangement) is the key to solving these problems effectively.