Forty teams play a tournament. Each team plays every other team just once. Each game results in a win for one team. If each team has a 50% chance of winning each game, the probability that at the end of the tournament, every team has won a different number of games is

Team totals must be 0, 1, 2, ..., 39. Let the teams be T1, T2, ..., T40, so that Ti loses to Tj for i < j. In other words, this order uniquely determines the result of every game. There are 40! such orders and 780 games, so 2780 possible outcomes for the games. Hence, the probability is 40!/2780.