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Gmat Permuation and Combination problem

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by lakshmi86 » Sat Aug 07, 2010 3:02 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


This is solved in the following link: https://gmatclub.com/forum/ps-playoff-time-1545.html

Please help me understand... why it has to be one minus in the first part for each time and why not no minus at all?
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by lakshmi86 » Sat Aug 07, 2010 7:19 am
Please help?
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by fitzgerald23 » Sat Aug 07, 2010 8:59 am
In this case you are maximizing losses within each division. The way to maximize the games played is to assume every team loses 2 games except the overall league champion that at most can lose 1 game.

Lets assume the champ comes from the 12 team league.

There will be a total of 18 losses in the 9 team league
There will be a total of 20 losses in the 10 team league
There will be a total of 22 losses in the 11 team league

That is a total of 60 games.

Now in the 12 team league 11 teams are going to lose twice, but the overall champion is only permitted one loss (it can lose one divisional game and still be overall champion. That is a total of 23 losses.

That makes an overall total of games to be played at 83 games.

You can write it out to visualize the scenario better:

Example 10 Team League

1 W L W W W W
2 W L W W L
3 W L W L
4 W L W L
5 W L W L
6 L W L
7 L W L
8 L W L
9 L W L
10 L W L

There are 5 games in each of the first three rounds before a team is eliminated. Two games and two eliminations in round 4. 1 Game and 1 elimination in round 5 and finally 1 game and 1 elimination in round 6. That is 19 total games in the division just to name the winner. You can do that for each division and the championship round to visualize the process better
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