The task is to assign 8 tennis players into pairs of 2 for a bracket.
The answer is 105, for which my textbook used the method: 1. Select the 4 pairs, for which there are 8C2*6C2*4C2*2C2 ways. 2. Divide by the number of permutations for the pairs, which is 4P4. (8C2*6C2*4C2*2C2)/4P4 = 105 pairings.
I'm wondering what I'm doing wrong with this other method that I tried: 1. Select all combinations of 4 players for the left side of the bracket, 8C4. 2. For each such combination, pair it with all permutations of the other 4 players, 4P4. 3. This should overestimate by a factor of 2 (mirror images), because the left-side {a,b,c,d} will duplicate with the left-side {e,f,g,h}.
However, (8C4*4P4)/2 = 840, which is still off by a factor of 8. What duplicates have a neglected to remove?