2. A tournament of the game "Rock, Paper, Scissors" ${ }^{3}$ is held according to the Olympic system, ${ }^{4}$ with 16 players registered. The organizers have determined who will play against whom in the first match, which winners of which matches will play against each other in the next, and so on. Each participant intends to throw the same thing in all their games, and what exactly that will be is chosen randomly before the very first game. How many ways do the players have to make their choices so that the tournament can conclude?
Solution. Let's consider the general case with $2^{n}$ players. To uniquely draw the tournament bracket, it is sufficient to know the figure of the tournament winner and whether the right or left player won in each of the $2^{n}-1$ matches. Indeed, then the figures of the winners of the left and right semi-finals are uniquely determined, from these figures - the figures of the winners of the quarter-finals, and so on, down to the initial arrangement.

It remains to note that different sets of data obviously lead to different brackets, so the answer is $3 \cdot 2^{2^{n}-1}$. In our specific case, $3 \cdot 2^{15}$.