Organizing a chess tournament

Six players are participating in a chess tournament. In the first phase of the tournament, each player plays against three other players. By double counting, there are 6×32=9 \frac{6 \times 3 }{2} = 9 games that are held.

How many different possibilities are there for the set of 9 different games to be played?

Details and assumptions

The order the games are played in does not matter, just which pairs of players compete.

As an explicit example, here is a possible set of 9 games: Player ii plays against player i1,i+1 i-1, i+1 and i+3 i + 3 , where calculations are done modulo 6.

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