Problem:
In a five-team tournament, each team plays one game with every other team. Each team has a chance of winning any game it plays. (There are no ties.) Let be the probability that the tournament will produce neither an undefeated team nor a winless team, where and are relatively prime positive integers. Find .
Solution:
The five teams must play a total of games, so there are possible outcomes for the tournament. Team wins all four of its games in of these outcomes. Because at most one team can be undefeated, there are tournaments that produce an undefeated team. A similar argument shows that of the possible tournaments produce a winless team. These possibilities are not mutually exclusive, however. In of the tournaments, team is undefeated and team is winless, and there are such two-team permutations. In other words, of the tournaments have both an undefeated team and a winless team. Thus, according to the inclusion-exclusion principle, there are tournament outcomes in which there is neither an undefeated nor a winless team. All outcomes are equally likely, hence the required probability is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions