Problem:
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a chance of winning any game it plays.The probability that no two teams win the same number of games is , where and are relatively prime positive integers. Find .
Solution:
Suppose that no two teams win the same number of games. Then, for any between and , exactly one team wins games. Moreover, a team that wins games can lose only to the team that wins all of its games. An inductive argument shows that in fact each team loses to any team that wins more games, but to no other teams. Thus a tournament in which no two teams win the same number of games is uniquely determined by listing the teams in order of their wins. Given any listing, the probability that each team beat all the teams below it and lost to all the teams above it is , where is the total number of games. Because there are ways to list the teams, the requested probability is . The fraction is not in lowest terms, however. The number of factors of in is
Thus the requested probability is , where is an odd integer, so .
The problems on this page are the property of the MAA's American Mathematics Competitions