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by MBAsa » Mon Jun 24, 2013 5:47 am
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
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by Brent@GMATPrepNow » Mon Jun 24, 2013 6:22 am
MBAsa 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


Phew, that's a long question! Probably a little too long for the GMAT quant section. That said, here's one approach.

In every game, there's 1 win and 1 loss. So, let's just count the losses.

Each division held its own double-elimination tournament
In a certain division, the maximum games will occur when all but one team loses 2 games AND the top team loses 1 game.
Let's refer to as the best team in a division as the TOP team, and refer to eliminated teams as LOSING teams.

So, in the 9-team division, we have 8 LOSING teams and 1 TOP team.
In the 10-team division, we have 9 LOSING teams and 1 TOP team.
In the 11-team division, we have 10 LOSING teams and 1 TOP team.
In the 12-team division, we have 11 LOSING teams and 1 TOP team.

So, in the first round there were 38 LOSING teams altogether, so they lost a total of 76 games (since these LOSING teams lost 2 games each)
Also, in the first round there were 4 TOP teams altogether, so they lost a total of 4 games (since they lost 1 game each)

So, in the first round, there were 76 + 4 games (80 games) altogether.


In the next round there are 4 TOP teams remaining.
With each game, 1 team is eliminated (since this is a single-elimination tournament)
So, to eliminate 3 of the teams (and thus determine a champion), we must play 3 games.

So, the maximum number of games = 80 + 3 = [spoiler]83 = B[/spoiler]

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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