Question Solved1 Answer 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.) 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 \( E \) ?

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Transcribed Image Text: 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.) 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 \( E \) ?
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Transcribed Image Text: 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.) 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 \( E \) ?
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