In this section, the following topics are discussed with examples

• Introduction
• Fundamental Principle of Counting
• Multiplication Principle
• Examples

### Introduction

Permutation is the different arrangement of a given number of things by taking some or all at a time and on the other hand combinations are the selection which can be done by taking some or all from a number of objects. This chapter is considered a tricky one by aspirants and hence, a clear understanding of concept is required, usually 2–3 questions are asked in any exam either directly or indirectly as application in probability.

### FUNDAMENTAL PRINCIPLE OF COUNTING

#### Multiplication Principle

Let’s take that if a process A can be performed in ‘a’ different ways; followed by a second process B is performed in ‘b’ different ways, then the two processes in succession can be performed in × ways.

## The only place where you will find success before work is in the dictionary.

May V Smith

#### Example 1:

A person wants to go from city to city via city B. There are 4 routes from to and 5 routes from to C. In how many ways can he travel from to

Solution:

So now lets see some basics of fundamental counting and how it works. He can go from to in 4 ways and B to in 5 ways.

So number of ways of travel from to is 4 × 5 = 20 ways.

#### Example 2:

In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes ?

Solution:

First prize may be given to any one of the 4 boys, hence rst prize can be distributed in 4 ways.

Similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways.

∴ The number of ways of their distribution

= 4 × 4 × 4 × 4 × 4 = 45 = 1024

If an operation can be performed in ‘m’ different ways and another operation, which is independent of the rst operation, can be performed in ‘n’ different ways. Then either of the two operations can be performed in (n) ways. This can be extended to any nite number of independent operations.

#### Example 3:

A college offers 6 courses in the morning and 4 in the evening. Find the number of ways a student can select exactly one course, either in the morning or in the evening.

Solution:

The college has 6 courses in the morning out of which the student can select one course in 6 ways.

In the evening the college has 4 courses out of which the student can select one in 4 ways.

Hence the required number of ways = 6 + 4 = 10.