Combinations in tournament

Question

A tournament includes P total players. The game played in rounds with teams of size T. Possible number of teams is (P T).

Questions

• How would you calculate the number of total possible combinations of matches?
• How would you calculate the number of total possible matches per round?

Examples

Example 1: 4 players in teams of 2
4 Players
1-2
1-3
1-4
2-3
2-4
3-4
=(4 2) = 6

1-2 - 3-4
1-3 - 2-4
1-4 - 2-3
= 3


Example2: 5 players in teams of 2
Teams
1-2
1-3
1-4
1-5
2-3
2-4
2-5
3-4
3-5
4-5
=(5 2) = 10

Matches
1-2 - 3-4
1-2 - 3-5
1-2 - 4-5
1-3 - 2-4
1-3 - 2-5
1-3 - 4-5
1-4 - 2-3
1-4 - 2-5
1-4 - 3-5
1-5 - 2-3
1-5 - 2-4
1-5 - 3-4
2-3 - 4-5
2-4 - 3-5
2-5 - 3-4
= 15

Answer 1

you have to chose the first and the second teams. for the first team you have $\binom{P}T$ and for the second team you have $\binom{P-T}T$ however you are counting each match twice in this case so the total number of possible matches is $$\frac{\binom{P}T\binom{P-T}T}{2}$$

The number of possible combinations of matches per round (assuming you get as many teams to play as possible in each round) is $$\prod_{n=0}^{n=2\lfloor{T/2P}\rfloor-1}\binom{P-(Tn)}T/(\lfloor{T/2P}\rfloor!(2^{\lfloor{T/2P}\rfloor})$$