Suppose four teams, numbered one through four, play a single-elimination tournament, consisting of three games. Two teams play each game and one of them wins; ties do not occur. The tournament bracket is as follows: teams one and another team play each other in the first game and the remaining two teams play each other in the second game; the winner of the first game plays the winner of the second game in the third game.
Define a set ΩΩ so the elements of ΩΩ correspond to the possible outcomes of the tournament. An element of ΩΩspecifies the entire sequence of outcomes of the games. How many outcomes are there for the combination of what bracket is used and the game outcomes? (Assume the order the two games are played in the first round does not matter. For example, they could be simultaneous.)