Fundamental Principle of Counting

Suppose two events E₁ and E₂ are to be performed in sequence, then if E₁ can be performed in 'm' ways and for each of these ways E₂ can be performed in 'n' ways, then the sequence E₁E₂ can be performed in 'mn' different ways. This is known as the Fundamental Principle of Counting.

Multiplication Principle:

The same principle can be generalized to three or more events occurring in succession as follows:

If n operations can be performed in m₁,m₂,m₃,...m_n ways respectively, then all n operations in succession can be performed exactly in m₁,m₂,m₃,...m_n ways.

The above principle is called multiplication principle.

Addition Principle:

If two events E₁ and E₂ can occur independently in exactly m ways and n ways respectively, then either of the two events can occur in (m + n) ways.

Example 1:

There are 8 cars plying between two towns A and B. In how many ways can a person go from one town to the other and return by a different car?

Suggested answer:

There are 8 ways of travelling from A to B and seven ways for the return journey, since he cannot come by the same car. Hence, the number of ways of making both the journeys is 8 x 7 = 56.

Example 2:

How many numbers are there between 100 and 1000 such that every digit is either 2 or 9?

Suggested answer:

Number of ways of filling hundred's place = 2

Number of ways of filling ten's place = 2

Number of ways of filling unit's place = 2

By the fundamental principle of counting, the total number of numbers = 2 x 2 x 2 = 8.