Fundamental principles of counting

. The Sum Rule Let us consider two tasks which need to be completed. If the first task can be completed in M different ways and the second in N different ways, and if these cannot be performed simultaneously, then there are M + N ways of doing either task. This is the sum rule of counting.

Example . Suppose one girl or one boy has to be selected for a competition from a class comprising boys and girls. In how many different ways can this selection be made? The first task of selecting a girl can be done in ways.

The second task of selecting a boy can be done in ways. It follows from the sum rule, that there are + = ways of making this selection. The sum rule may be extended to more than two tasks. Thus if there are n non-simultaneous tasks T , T , T , · · · , T n which can be performed in m , m , · · · , m n ways respectively, then the number of ways of doing one of these tasks is m + m + · · · + m n .

. The Product Rule Let us suppose that a task comprises of two procedures. If the first procedure can be completed in M different ways and the second procedure can be done in N different ways after the first procedure is done, then the total number of ways of completing the task is M × N Example . Consider the cities Chennai, Trichy and Tirunelveli.

In order to reach Tirunelveli from Chennai, one has to pass through Trichy. There are roads connecting Chennai with Trichy and there are roads connecting Trichy with Tirunelveli. What are the total number of ways of travelling from Chennai to Tirunelveli? .

Fundamental principles of counting There are roads connecting Chennai to Trichy. Suppose these are R and R . Further there are roads connecting Trichy to Tirunelveli . Let us name them as S , S and S .

Suppose a person chooses R to travel from Chennai to Trichy and may further choose any of the roads S , S or S to travel from Trichy to Tirunelveli. Thus the possible road choices are ( R , S ) , ( R , S ) , ( R , S ) . Similarly, if the person chooses R to travel from Chennai to Trichy, the choices would be ( R , S ) , ( R , S ) , ( R , S ) .