Organizing A Tournament
You are organizing a soccer tournament. Each team plays against exactly 3 distinct teams.
In the U-18 group, you have 6 teams and you are required to make up a schedule. One possibility is:
1 vs 2, 1 vs 3, 1 vs 4, 2 vs 5, 2 vs 6, 3 vs 5, 3 vs 6, 4 vs 5 and 4 vs 6.
How many different schedules can you make?
- The order in which a team plays it's game did not matter. What matters is the opponents a particular team played.
- The teams are not identical.
The answer is 70.
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1 solution
Let us call the teams 1, 2, 3, 4, 5 and 6.
Choose team 1`s opponents. You have 10 ways of choosing 3 teams from 5 teams.
Let us say 1 vs 2, 1 vs 3 and 1 vs 4.
case A: if teams 1`s opponents don't play against each other. If follows that each of 2, 3 and 4 have to play 5 and 6. (5 and 6 can't play each other because either 2, 3 or 4 would not have 3 opponents). We then have exactly 10 ways.
case B: Some of teams 1`s opponent play each other. Exactly one of the following can happen 2 v 3 or 2 vs 4 or 3 vs 4 ( 3 choices).
You can`t have 2 of (2 vs 3, 2 vs 4, 3 vs 4) happening because teams 5 or 6 would not have enough opponents. Let us say 2 vs 3 then team 2 must choose either 5 or 6 ( 2 choices) and everything else follows. We have 10 x 3 x 2 = 60 ways.
The total number of ways is 10 + 60 = 70 ways