M43-38

Question

There are  n  teams in Group A, and each team in Group A played each of the other teams exactly  n  times. What was the total number of games played by the teams in Group A?

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
 
(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

Bunuel · Answer

Official Solution:

There are  n  teams in Group A, and each team in Group A played each of the other teams exactly  n  times. What was the total number of games played by the teams in Group A? 
 
There are  C^2_n  pairs possible from  n  teams, and since each pair played  n  times, the total number of games played is:
   
 C^2_n*n= 
 
 =\frac{n!}{2!(n-2)!}*n= 
 
 =\frac{(n-2)!(n-1)n}{2(n-2)!}*n= 
 
 =\frac{(n-1)n}{2}*n= 
 
 =\frac{(n-1)n^2}{2} .
   
(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
 
Each team played  (n - 1)  other teams  n  times, hence each team played  (n - 1)n  games. Hence, this statement implies that  10 < (n-1)n < 25 . By trial and error, we can find that two values of  n  satisfy this: 4 and 5. Not sufficient.
 
(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.
 
This statement implies that:
   
 2*(n - 1)n=\frac{(n-1)n^2}{2} 
 
 2=\frac{n}{2} 
 
 n=4 
   
Sufficient.

Answer: B

lnyngayan · Answer

Could you further explain why it is (n-1)*n games in total? 
Many thanks!

Bunuel · Answer

Each team plays against (n-1) other teams because there are n teams in total, and one team cannot play against itself. Since each team plays every other team n times, the total number of games played by each team is (n-1) * n.

Let's say there are 4 teams: A, B, C, and D. Each team plays against the other 3 teams. For example:

Team A plays B, C, and D, so A plays 3 games. 
 Team B plays A, C, and D, also 3 games, and so on for the others.

If each team plays every other team 4 times, then each team plays 3 * 4 = 12 games.

In total, every team plays (n-1) * n games, where n is the number of teams.

MaudD · Answer

I think this is a  poor-quality  question and I don't agree with the explanation. Solution (2) supposes that the number of teams is at least two. Without this hypothesis, the solutions of 0 team or 1 team in group A are possible options. 
The question should say that n >= 2.

Bunuel · Answer

I don't agree. This is not a poor-quality question, and the explanation is correct. The problem inherently requires at least two teams, as each team is meant to play against the others. A scenario with 0 or 1 team wouldn’t make sense because no games could be played in such cases.

MaudD · Answer

Apologies, my previous message got automatically generated from the "send feedback" button. I do think this is an excellent question, and that the answer is elegant. However I disagree with the implicit hypothesis that if the question mentions a Group A where each team plays against each of the other teams, there must be at least two teams in that group A.
From a pure mathematical point of view, having 0 or 1 teams in group A are twisted solutions, but they are legit to me.
So I still disagree with the inherence of having at least two teams, from the way the question is formulated.
But anyways this question is good training!

Bunuel · Answer

I see where you're coming from, but I still disagree. The cases need to be realistic. For example, if there were 0 or 1 team in Group A, the statement "the number of games played by each team in Group A is half the total number of games played by all the teams in Group A" wouldn’t make sense, as there would be no games to discuss in the first place.

bbucey · Answer

Is (2) supposed to be

(n−1)n=(n−1)n^2/2 instead? Why do we multiply by 2 in (2) but not in (1)? I thought that we established in (1) that the 'number of games played by each team in Group A' was n(n-1)?

Bunuel · Answer

(2) says: The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

The number of games played by each team in Group A is  (n - 1)n , the same as in (1).

The total number of games played by all the teams in Group A is  \frac{(n-1)n^2}{2} .

We are told that the first number,  (n - 1)n , is half of the total,  \frac{(n-1)n^2}{2} , so we get  2*(n - 1)n = \frac{(n-1)n^2}{2} .

Hope it's clear.

CosmicWanderer · Answer

If the number of games played by each team in Group A is (n−1)n, shouldn't the number of games played by all teams be (n−1)n^2 ?

Bunuel · Answer

No, it's (n−1)n^2/2 because a game played between two teams, say Team A and Team B, is counted twice: once in the count of games for Team A and once in the count of games for Team B. So, (n−1)n^2 includes duplicate games—twice as many. To get the correct total number of games, we divide that by 2.

CosmicWanderer · Answer

Thankyou

GMATPrecision · Answer

Bunuel - can't the question be interpreted this way?

Bunuel · Answer

I can’t really see what you wrote there. I’ll just go ahead and say no, because the wording of the question is precise and the official interpretation is correct.

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