Why do these seemingly logically equivalent combinations give different answers? I'm having trouble figuring out why these two different ways to write this combination give different answers. Here is the scenario:
Q: Choose a group of 10 people from 17 men and 15 women, in how many ways are at most 2 women chosen?
Solution A: From 17 men choose 8, and from 15 women choose 2. Or from 17 men choose 9, and from 15 women choose 1. Or from 17 men choose 10.
C(17,8)*C(15,2)+C(17,9)*C(15,1)+C(17,10) = 2936648 ways
Solution B: Choose from the men to fill the first 8 positions and choose the next 2 positions from the remaining men and women.
C(17,8)*[C(9,2)+C(9,1)*C(14,1)+C(14,2)] = 6150430 ways
What is wrong with my logic or interpretation here? 

Consider a simpler problem. How many ways are there to choose ten men from a set of ten men? By your second reasoning, choose eight first, then choose another two. That gives a total of:
$$\binom{10}{8}\binom{2}{2}=45$$
Why is that reasoning problematic?
