AkamaiBrah wrote:
A certain league has four divisions. The respective divisions had 9, 10, 11, and 12 teams qualify for the playoffs. Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games -- in order to determine its champion. The four division champions then played in a single-elimination tournament -- where a team is eliminated upon losing one game -- in order to determine the overall league champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?
(A) 79
(B) 83
(C) 85
(D) 87
(E) 88
Let’s call the four divisions A, B, C, and D, with 9, 10, 11, and 12 teams, respectively. Now let’s analyze the maximum number of games that can be played in each division.
Let’s take division A, for example. It has 9 teams and only 1 is the winning team. So, there must be 8 losing teams, and each of these teams must lose twice since it’s a double-elimination tournament. The winning team can still lose once (but not twice; otherwise it’s out of the tournament). Therefore, the maximum number of games played in division A is 8 x 2 + 1 = 17.
Notice that 8 = 9 - 1; thus, using the same argument, the maximum number of games played in each remaining division is:
Division B: (10 - 1) x 2 + 1 = 18 + 1 = 19
Division C: (11 - 1) x 2 + 1 = 20 + 1 = 21
Division D: (12 - 1) x 2 + 1 = 22 + 1 = 23
Thus, the maximum total number of games played in the 4 divisions before the single-elimination is 17 + 19 + 21 + 23 = 80.
In the single-elimination, only 3 games will be played since there will be 2 semi-finals and 1 final. (For example, in the semi-final 1, A’s champion vs. B’s champion; in the semi-final 2, C’s champion vs. D’s champion; and in the final, the winner between A and B takes on the winner between C and D to determine the overall league champion.)
Thus, the maximum total number of games played to determine the overall league champion is 80 + 3 = 83.
Answer: B