2013 AMC 10A Problems/Problem 24

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

Solution

• Credit to the Math Jam for this solution

Let us label the players of the first team , , and , and those of the second team, , , and .

Let us first consider how to organize A's matches, , , , , , and . Because we have three duplicates, there are ways to organize A's matches.

Now, consider and . WLOG assume that A's matches were , as we will multiply by the end anyways, and that, in the first round, played and played .

There are two cases.

1. plays again in the second round (and plays in the second round)

In this case, the rest of the matches are forced, as must play in both of rounds and (as it has already played twice) and same with rounds and with and . Thus, there is only one option.

2. plays in the second round (and plays in the second round)

In this case, can play in either round or and in either round or , so there are options.

Thus, with playing in the first round, there are options. Multiplying this by for the case where plays in the first round, we get options.

Finally, to get our final answer, we multiply ways to organize the matches.