Let the players be numbered 1 to 20.

First we think of there being two teams, Team A and Team B

We choose the players for team A in C(20,10) ways, or 184756, 
and the remaining 10 will be on team B.

But this is too many because, for instance, the 184756 counts

Team A = {2,5,6,9,11,14,15,16,18,20)
Team B = {1,3,4,7,8,10,12,13,17,19}

and this

Team A = {1,3,4,7,8,10,12,13,17,19}
Team B = {2,5,6,9,11,14,15,16,18,20)

as 2 different choices, but they are not.  Therefore we
must divide the 184756 by 2!

Answer: 92378

Edwin