Question 3: Consider a single elimination tournament of 16 football teams, laid out as in the...

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Question 3: Consider a single elimination tournament of 16 football teams, laid out as in the diagram below. Once a team loses it is out of the tournament. Every team must play until it is eliminated. The starting positions for each team are fixed. A matchup (X,Y) represents a game between teams X and Y. A round is the set of all the matchups at a given stage of the tournament (the matchups above the round title in the diagram). Thus round 3 in the diagram below is the set {(D, E), (I,O)} of matchups. An outcome is the union of all the matchups along with the winner. (Essentially two outcomes are different if the winners of the games produce a different letter in at least one place in the diagram below.) AB A A B C I L -K -L E E WINNER N -M F E G G H Round 1 Round 2 Round 3 Round 4 Round 3 Round 2 Round 1 a) How many unique outcomes are there in this tournament? b) How many unique outcomes are there where A wins the entire tournament? c) How many unique outcomes are there where D never faces E? Question 3: Consider a single elimination tournament of 16 football teams, laid out as in the diagram below. Once a team loses it is out of the tournament. Every team must play until it is eliminated. The starting positions for each team are fixed. A matchup (X,Y) represents a game between teams X and Y. A round is the set of all the matchups at a given stage of the tournament (the matchups above the round title in the diagram). Thus round 3 in the diagram below is the set {(D, E), (I,O)} of matchups. An outcome is the union of all the matchups along with the winner. (Essentially two outcomes are different if the winners of the games produce a different letter in at least one place in the diagram below.) AB A A B C I L -K -L E E WINNER N -M F E G G H Round 1 Round 2 Round 3 Round 4 Round 3 Round 2 Round 1 a) How many unique outcomes are there in this tournament? b) How many unique outcomes are there where A wins the entire tournament? c) How many unique outcomes are there where D never faces E?