Formula for finding an unknown value

[eselpungen]

Hello everyone.

I'm having an issue where i need to create a formula for the following scenario/criteria:

There are 10 (n) footballplayers which are put on 2 (X) teams (5 on each team).
For each match they play against each other the teams is changed.
The same team should not occour more than once.
How many matches can they play before every possible team has been played?

I'm not really looking for the answer to this particular scenario, more a formula for calculating it.
A formula where we know how many players, and how many teams, where the result gives me the number of matches.

Feel free to move this thread if it belongs in a more approprite category.

Any help is MUCH appretiated as I'm really stuck on how to solve this.

Best regards, Harald

 

[pka]

There are 10 (n) footballplayers which are put on 2 (X) teams (5 on each team).
For each match they play against each other the teams is changed.
The same team should not occour more than once.
How many matches can they play before every possible team has been played?
I'm not really looking for the answer

It is hard to not give you the answer. These are unordered partitions.
If M is a multiple of K, \(\displaystyle M=N\cdot K\), then the number of ways to form \(\displaystyle N\) cells of \(\displaystyle K\) each from those \(\displaystyle M\) distinct objects is \(\displaystyle \dfrac{M!}{(K!)^N(N!)}\).

 

[eselpungen]

Thank you so much for your reply!
I had to read up on unordered partitions and factorials, but got it in the end!

Once again, thank you for your (really fast!) reply, and merry christmas!