(a) 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. In the first round, teams one and two play each other in the first game and teams three and four play each other in the second game. In the second round, the winner of the first game plays the winner of the second game in the third game. Define a set so that the elements of correspond to the possible outcomes of the tournament. An element of should specify the entire sequence of outcomes of the games. Explain how the elements of your set correspond to the possible outcomes. (This is an exercise in coming up with good notation.) (a) where indicates which of the two teams playing the game wins; indicates that the higher numbered team wins and indicates that the lower numbered team wins. (b) where is the number of games that team wins in the tournament. (c) is the set of permutations of , where a permutation indicates the ranks of the four teams based on their performance in the tournament. Select all possible options that apply. (b) How may possible outcomes are there