### Instruction: Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$.### Response: I notice that the problem is asking about the probability of a very unlikely event: that no two teams win the same number of games.
This means that the fraction $\frac mn$ will be very small, and therefore $n$ will be very large.
In fact, $n$ will be the total number of possible outcomes for the tournament, since every outcome is equally likely.
So I want to find out how many possible outcomes there are for the tournament.
I realize that each game has two possible outcomes: either one team wins or the other team wins.
There are $\binom{40}{2} = 780$ games in the tournament, since every pair of teams plays once.
Therefore, the total number of possible outcomes for the tournament is $2^{780}$.
This means that $n = 2^{780}$, and $\log_2 n = 780$.