# Basic Principles of Counting

## Basic Principles of Counting:

There are two fundamental principles connected to the process of counting. These two principles are known as the principle of addition and the principle of multiplication.

The Fundamental Principle of Addition states, “If two operations can be performed independently in m and n ways respectively, then the total number of ways in which either of these two operations can be performed is (m + n).”

Let us take one simple example to make the point clear. Consider that there are 5 candidates for a prize in debate, 7 candidates for a prize in recitation, and 3 candidates for a prize in music. In how many ways can one of these prizes be given? In this case, the prize for debate can be given to any one of the 5 candidates. So, this job of awarding the prize for debate can be done in 5 ways. Similarly, the second job of awarding the prize for recitation can be performed in 7 ways. The third job can be performed similarly in 3 ways. Obviously, then, the number of ways in which any one of these three jobs can be performed in (5 + 7 + 3) i.e. 15. Remember that in this case, the aim is to perform either the first job or the second job, or the third job.

The Fundamental Principle of Multiplication states, “If one operation can be performed in m ways and if after the first operation has been performed in any one of these ways, a second operation can be performed in n ways, then the number of ways in which these two operations can be performed together is m x n?”

In the above example if the question is to find out the number of ways in which the prizes in the three categories can be awarded, then our line of thinking would be different. Here, all the three prizes are to be awarded. So the question is to perform each of the three jobs (and not any one of them). The first job can be performed in 5 ways. When the first job has been performed in any one of the possible 5 ways, the second job can be performed in 7 ways. So, the number of ways of performing these two jobs together is 5 x 7. When these two jobs have been performed in any one of the possible 35 ways, the third job can be done in any one of 3 ways. So, the number of ways in which these three jobs can be performed together is 35 x 3 i.e. 105.

The basic difference between the principle of addition and the principle of multiplication is that in the former case, any one of the different jobs is to be performed, whereas, in the latter, each of the different jobs is to be performed. The fundamental principles of addition and multiplication can be extended to any number of operations.