Oh, you want to make things complicated, do you?
I'm up for the challenge. Let's say the Jazz have 3 PG (Harris, Tinsley, Watson), 5 "Wings" (Bell, Burks, Hayward, Howard, and Miles), and 5 "Bigs" (Evans, Favors, Jefferson, Kanter, and Millsap), and that Corbin has to pick 1 PG, 2 Wings, and 2 Bigs for each lineup.
In that case, he has exactly 300 different lineups he can choose from. ( 300 = (3 choose 1) * (5 choose 2) * (5 choose 2) , where "a choose b" = a!/(b!(a-b)!) )
One could make things even more complicated, by saying for example Evans can play SF or PF, but not SG or C. In that case you get only 210 different lineups. (Not going to show work for this one unless anyone really cares.)
Boo-yah!
Oh, come on... *someone's* got to give me rep for this!