An ordinary knock-out singles tennis tournament (with no seeds or byes) consists of a series of rounds. In each round, the remaining players play against each other in pairs, with the losers being eliminated and the winners going through to the next round; in the last round, the only two remaining players play the final match to determine the winner of the tournament.
Suppose that there are 7 rounds. Before the tournament starts, the organizers need to construct the draw, which specifies who plays who in the first round, and then which first-round winners play which other first-round winners in the second round, and so on. Of course, the organizers don’t know who the first-round winners will be, so in the draw they are just thought of as “the winner of the match between player X and player Y”, and so forth. How many possible draws are there? Use the fact, proved in lectures, that the number of ways to group 2k people into k pairs is $\frac{(2k)!}{(2^k)k!}$
