Problem: Some students in a gym class are wearing blue jerseys, and the rest are wearing red jerseys. There are exactly 25 ways to pick a team of three players that includes at least one player wearing each color. Compute the number of students in the class.
My failed solution: When I tried to do this question, I defined $r$ as being the number of players on the red team, and $b$ as being the number players on the blue team. Thus the total number of ways to pick a team of three would be $((b)(r)(b+r-2))/(3!)=25$. This does not lead to a solution for $b$ or $r$ (using prime factorization). What did I do wrong?
It's okay to say pick a blue guy, then a red guy, then anybody who's left. The only mistake is in how you adjusted for double counting. You've only counted each team twice, not six times. Say Alex and Bob are blue and Charlie is red. In you scheme, you count Alex, Charlie, Bob and Bob, Charlie, Alex, but no other permutation.
You should have gotten $$br(b+r-2)=50$$