FUNDAMENTAL PRINCIPAL OF COUNTING

Rule of Product

If one experiment has n possible outcomes and another experiment has m possible outcomes, then there are m × n possible outcomes when both of these experiments are performed.

In other words if a job has n parts and the job will be completed only when each part is completed and the first part can be completed in a₁ ways, the second part can be completed in a₂ ways and so on ...... then nth part can be completed in an ways, then the total number of ways of doing the job is a₁a₂a₃ ... a_n. This is known as the rule of product.

Illustration:

A college offers 7 courses in the morning and 5 in the evening. Find the possible number of choices with the student who wants to study one course in the morning and one in the evening.

Solution:

The student has seven choices from the morning courses out of which he can select one course in 7 ways.

For the evening course, he has 5 choices out of which he can select one in 5 ways.

Hence the total number of ways in which he can make the choice of one course in the morning and one in the evening = 7 × 5 = 35.

Illustration:

A man has five friends. In how many ways can be invite one or more of them to a tea party.

Solution:

Methods I

Case I:     The man can invite one friend in ⁵C₁ ways

Case II:    The man can invite two friend in ⁵C₂ ways

Case III:   The man can invite three friend in ⁵C₃ ways

Case IV:   The man can invite four friend in ⁵C₄ ways

Case V:    The man can invite five friend in ⁵C₅ ways

All these five cases are mutually exclusive i.e. the man can either invite one friend or two friends or ............., but can not simultaneously invite one friend and two friends.

Method II.

This problem can also be solved as follows:

Each of his friends may be dealt in two ways - either invited or not invited.

Therefore  The total number of ways of inviting all friends is

                = 2 × 2 × 2 × 2 × 2 = 32 ways.

But this includes the case, in which all the friends are not invited. Hence, the total number of ways in which the man invites one or more friends is 25 - 1 = 31 ways.

Note:  In the above illustration, we have found the total number of combinations of n dissimilar things taking any number of them at a time, which is 2n - 1.