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

Question

Infinity Learn · Accepted Answer

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 $$40$$!$$/2780$$.

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

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