Elimination Tournaments In an elimination tournament the teams are arranged in opponent pairs for the first round, and the winner of each round goes on to the next round until the champion emerges. The following diagram illustrates a 16-team tournament bracket, in which the 16 participating teams are arranged on the left under Round 1 and the winners of each round are added as the tournament progresses. The top team in each game is considered the "home" team, so the top-to-bottom order matters. Round I Round 2 Round 3 Round 4 (Quarter final) (Semifinal) (Final) Champion To seed a tournament means to select which teams will play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15. This exercise is based on various types of elimination tournaments. (Leave each answer as a formula. Do not enter your answers using the notation P(n,r) or C(n,r).) (a) How many different seedings of a 16-team tournament are possible? seedings (b) In how many seedings will the top-ranked team play the bottom-ranked team, the second-ranked team play the second-lowest-ranked team, and so on? seedings

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Elimination Tournaments In an elimination tournament the teams are arranged in opponent pairs for the first round, and the winner of each round goes on to the next round until the champion emerges. The following diagram illustrates a 16-team tournament bracket, in which the 16 participating teams are arranged on the left under Round 1 and the winners of each round are added as the tournament progresses. The top team in each game is considered the "home" team, so the top-to-bottom order matters.

A tournament bracket has four rounds. Starting with the round in the leftmost column and moving right, the rounds are as follows.

Round 1 has sixteen entries organized into pairs. Each pair leads to a single entry in round 2.
Round 2 (quarter final) has eight entries organized into pairs. Each pair leads to a single entry in round 3.
Round 3 (semifinal) has four entries organized into pairs. Each pair leads to a single entry in round 4.
Round 4 (final) has a single pair of entries. The pair leads to a single entry labeled "champion."

To seed a tournament means to select which teams will play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.† This exercise is based on various types of elimination tournaments. (Leave each answer as a formula. Do not enter your answers using the notation

P(n, r)

C(n, r).)

(a)

How many different seedings of a 16-team tournament are possible?

__ = seedings

(b)

In how many seedings will the top-ranked team play the bottom-ranked team, the second-ranked team play the second-lowest-ranked team, and so on?

___ = seedings

(ATTACHED IS THE IMAGE OF QUESTIONS, ANSWER (A) AND (B))