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AMC 10

AMC 10

I'm confused about the below problem, can some teacher help explain it in detail?

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?

Re: AMC 10

We can take a look at this problem in detail during the live session today. Please remember to let us know what you have tried so far so we can help you better.

In the meantime, here's a hint to get you started.

Say the players from Central High School are $A_1$, $A_2$, and $A_3$, and the players from Northern High School are $B_1$, $B_2$, and $B_3$. Let the players from CH choose their opponents. If $A_1$ chooses first they have $3$ players to choose from, then $A_2$ has $2$ players to choose from, and $A_3$ plays against the remaining player. Thus, there are $3! = 6$ ways to choose each round. Note this is the same as the number of rounds, and each player plays each player from the opposing team twice. Note the order of the rounds matters. Now, is this the only way to generate the $6$ rounds?