Given 11 chess players and 5 distinct tables, in how many ways can we pair them to play (color does matter)?
My problem is that I have found two approaches, both of which give different numbers, and I am not sure what is missing in one or double-counted in the other.
The first approach is to just take any permutation of the $11$ players, as the tables are distinct there are 11 unique spots (one of them is not playing), so we can just place the players according to the permutation. This gives $11!=39916800$ possible games.
The second approach is to first choose pairs, the first player has 10 choices, the next has 8, ... and then we multiply by $2^5$ to account for the colors of each pair, and finally multiply by the number of ways to seat them at tables (so multiply by $5!$), giving
My intuition is that the first approach is correct, but then I am not sure which pairings are missing in the second one..