EXAMPLE 35 Counting without repetition: Intramural singles tennis

A local college has an intramural singles tennis league with five players: Ryan, Megan, Nicole, Justin, and Kyle. The college presents a trophy to the top three players in the league. How many different possible sets of three trophy winners are there?

Solution

The major difference between Example 34 and this example is that, in this example there can be no repetition. Ryan cannot finish in first place and second place. So we proceed as follows: Five possible players could finish in first place, so . Now there are only four players left, one of whom will finish in second place, so . That leaves only three players, one of whom will finish in third place, giving . Thus, by the Multiplication Rule for Counting, the number of different possible sets of trophy winners is