Football team combinations

With 10 players remaining, the number of possible picks is $10*9*8*7*6$ =10 9 8 7 6. However, this result includes the same players in different orders, so we must divide the result by the number of ways the same people can be ordered ( $5*4*3*2*1$ =5 4 3 2 1), which results in 252. Originally, I made the mistake of then multiplying this by 2 to apply these combinations to both coaches, but there is no need for this: once one team is decided, the other is dictated implicitly, so as all the combinations are being applied to one team, the opposite players are automatically applied to the other.

There are 10 people left to be selected. Each captain will select 5 players. Although it might matter to the players what order they are selected in, it does not matter for the composition of the team. Thus, this is a combination . There are $\binom{10}{5}=\boxed{252}$ ways for Brandon to select his players, and then Meredith will have the rest.