How Many Ways Can This Competition Be Organized?

Oct 2013
713
91
New York, USA
A game has five people. Each round is one person against another. There are five rounds, and everyone competes against two other people. An example is 1 vs. 2, 3 vs. 4, 5 vs. 1, 2 vs. 3, and 4 vs. 5. Given these rules, how many ways can the competition be organized?

1. A person cannot compete against himself or herself.

2. A matchup cannot repeat.

3. Person 5 is new and cannot play in Round 1 so she can watch at least one round before playing.

4. Every round must be between two people competing for the first time or two people competing for the second time except for Round 3, which will have a person competing for the first time against a person competing for the second time.

5. The order of the pairings matter. For example, these are different ways:
1 vs. 2, 3 vs. 4, 5 vs. 1, 2 vs. 3, and 4 vs. 5
3 vs. 4, 1 vs. 2, 5 vs. 1, 2 vs. 3, and 4 vs. 5

6. The order of the two people within the pairing does not matter. 1 vs. 2 is the same as 2 vs. 1.
 

romsek

Math Team
Sep 2015
2,964
1,674
USA
Your description is too confusing to proceed.

what does "everyone competes against two other people" mean?
 
Oct 2013
713
91
New York, USA
There are five people and five one vs. one rounds. Each person competes twice with the times against different opponents. You can think about it as a group of five sports teams each having played two games without any pair of teams playing each other both times.

I did it, and I think there are 432 possibilities. For the Match 1, there are 10 combinations (5C2) of which 6 do not have Person 5. For Match 2, there are three people who haven't played yet, and 3C2 = 3. For Match 3, the one person who hasn't played faces any of the other 4. Matches 4 and 5 can be thought of together because Match 5 has whoever hasn't played twice yet. For Match 4, 4C2 = 6. 6*3*4*6 = 432.
 
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