Forty team 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 the end of the tournament, every team has won a different number of games is a.`1//780` b. `40 !//2^(780)` c. `40 !//2^(780)` d. none of these

Options

A. `1//780`
B. `40!//2^(780)`
C. `36//.^(64)C_(3)`
D. `98//.^(64)C_(3)`

Answer

B

Explanation

Team totals must be 0, 1, 2, …., 39. Let the teams be `T_(1)T_(2),…, T_(40)`, so that `T_(i)` loses to `T_(j)` for `i lt j`. In other words, this order uniquely determines the result of every game. There are `40!` such orders and 780 games, so `2^(780)` possible outcomes for the games. Hence, the probability is `40!//2^(780)`.