# Question Solved1 AnswerQuestion 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 $$E$$ ?