COUNTING PRINCIPLES

There are two fundamental counting principles viz. Multiplication principle and Addition principle.

Multiplication Principle:

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

Illustration 1:

A college offers 7 courses in the morning and 5 in the evening. Find the possible number of choices with the student if he  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 x 5 = 35.

Illustration 2:

A person wants to go from station A to station C via station B. There are three routes from A to B and four routes from B to C. In how many ways can he travel from A to C?

Solution:

A -> B in 3 ways

B -> C in 4 ways

=> A -> C in 3 x 4 = 12 ways

Remark:

The rule of product is applicable only when the number of ways of doing each part is independent of each other i.e. corresponding to any method of doing the first part, the other part can be done by any method.

If one experiment has n possible outcomes and another has m possible outcomes, then there are (m + n) possible outcomes when exactly one of these experiments is performed.

In other words, if a job can be done by n different methods and for the first method there are a1 ways, for the second method there are a2 ways and so on . . . for the nth method, an ways, then the number of ways to get the job done is (a1 + a2 + … + an).

Illustration 4:

A college offers 7 courses in the morning and 5 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 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 course in 5 ways.

Hence he has total number of 7 + 5 = 12 choices.