Three schools have a chess tournament. Four players come from each school. Each player plays three games against each player from the other schools, and plays one game against each other player from his or her own school. How many games of chess are played?

mathbum  Sep 16, 2018

There are 4C2 = 6 ways to select 2 schools.

Given those 2 schools there are 42 = 16 ways to pair up players 1 from each school

There are 3 games played by each of those 3 pairs.


This is a total of 6 x 16 x 3 = 288 games played against other schools.


There are 4C2 = 6 ways to select a pair from the 4 people at a given school.

There are 4 schools

They play 1 game each


This is a total of 6 x 4 x 1 = 24 games played against schoolmates


A total of 288 + 24 = 312 games

Rom  Sep 16, 2018

Hey Rom, wouldn't it be 18 games against schoolmates? There are 3 schools not 4.

mathbum  Sep 17, 2018

Also, I think you doubled the answer, right? 4 students from School A competing with 4 students from School B is the same as 4 students from School B competing with 4 students from School A. Shouldn't the answer be 162?

mathbum  Sep 17, 2018

I got the same answer (162). I too thought that rom doubled the number of games but now i see that his solution is correct, he just answered the question for 4 schools instead of 3.

Guest Sep 17, 2018

Rom: Have you seen this question? 



Is there a way of solving it, using permutations, combinations, factorials, partitions....etc. other than brute-force computer tabulation of each combination, as is done here in the solution appended to the question? I have tried and failed miserably!.

Thanks for looking at it.

Guest Sep 16, 2018

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