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Chapter 7: Problem 42

Eight players, \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}\), and \(\mathrm{H}\), are com- peting in a series of elimination matches of a tennis tournament in which the winner of each preliminary match will advance to the semifinals and the winners of the semifinals will advance to the finals. An outline of the scheduled matches follows. Describe a sample space listing the possible participants in the finals.

Short Answer

Expert verified
The possible participants in the finals of the tennis tournament are: A vs. E, A vs. F, A vs. G, A vs. H, B vs. E, B vs. F, B vs. G, B vs. H, C vs. E, C vs. F, C vs. G, C vs. H, D vs. E, D vs. F, D vs. G, and D vs. H, resulting in a total of 16 combinations.

Step by step solution

01

Understand the schedule of matches

We can represent the schedule of the tournament as follows: First, we have preliminary matches, where the winners will advance to the semifinals, and then the winners of the semifinals will advance to the finals. We can label the preliminary matches as X, Y, and Z: X has players A and B competing, Y has players C and D competing, and Z has players E, F, G, and H competing. Next, we have the semifinals: the winner of X will play against the winner of Y, and the winner of Z will play the last semifinal match. Finally, the winners of the semifinals will compete in the finals.
02

Determine the winners of the preliminary matches

We can list all the possible winners of the preliminary matches: Match X: A or B (winner: player A or B) Match Y: C or D (winner: player C or D) Match Z: E, F, G or H (winner: player E, F, G, or H)
03

Determine the winners of the semifinals

We can now determine all the possible winners of the semifinals, based on the preliminary match results: Semifinal 1 (winner of X vs. winner of Y): {A, B} vs. {C, D} Semifinal 2 (winner of Z vs. last semifinal match): {E, F, G, H} vs. {} Since the players in the second semifinal are unknown, we cannot determine the exact combinations for the semifinals. Nonetheless, based on the possible winners of the preliminary matches, we can pair the possible winners of semifinal 1.
04

Possible finalists

Finally, we can list all the possible combinations of final match participants based on the possible winners of the semifinals: - A vs. E - A vs. F - A vs. G - A vs. H - B vs. E - B vs. F - B vs. G - B vs. H - C vs. E - C vs. F - C vs. G - C vs. H - D vs. E - D vs. F - D vs. G - D vs. H These 16 combinations represent all possible participants in the finals of the tennis tournament.

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