Fundamental Principle of Counting

The fundamental principle of counting** **deals with the counting of sample points in a sample space. Also known as the multiplication rule for choices, it is stated as follows:

If an operation can be performed in n_{1} ways, and for each of these a second operation can be performed in n_{2} ways, and for each of the latter a third operation can be performed in n_{3} ways,......, and for each of the latter a kth operation can be performed in n_{k} ways, then the entire sequence of k operations can be performed in n_{1} n_{2} n_{3}.... n_{k} ways.

EX. Sue goes shopping for an outfit. She buys five blouses, three skirts and two pairs of shoes. For each of the five blouses, she can wear three skirts, and for each combination of blouse and skirt, she can put on two pairs of shoes. Thus, there are a total of (5)(3)(2) = 30 possible outfits than she can put together.

EX. An executive council is to be formed from a pool of 10 qualified candidates. The council will be composed of a Chairman, Vice-Chairman, Secretary and Treasurer. All 10 candidates can be appointed for any positions.

Any of the 10 candidates can be appointed as Chairman. Then, any of the remaining 9 candidates can be appointed as Vice-Chairman, the remaining 8 for Secretary, and the remaining 7 for Treasurer. Thus, there are (10)(9)(8)(7) = 5040 committees that can be formed.