Somehow, I am getting a different answer. What is wrong here?
2*(7C2*5C4) + 2*(7C3*5C3) = 210 + 700 = 910
Thats a good question and I understand what you are saying.
what you are doing is taking out one man at a time but doing so repeats some of the team that you form.
suppose m1, m2, m3, m4, m5, m6, m7, and m8 are the members of 8 men. Of them, m1 and m2 cannot be in the team. Lets consider each condition.
Condition 1: Taking out m1
Remaining members are: m2, m3, m4, m5, m6, m7, and m8, we need to combine 2 out of these 7 members. the possibilities is 7c2. fine.
Condition 2: Taking out m2
Remaining members are: m1, m3, m4, m5, m6, m7, and m8. Again we need to combine 2 men out of 7. The possibilities are same i.e. 7c2. Here the problem arises. In condition 1 you have, for example, a team of 2 consisting m3 and m4 that is repeted here as well. This way any combination from "m3, m4, m5, m6, m7, and m8" is repeted.
Therefore we cannot do => "2*(7C2*5C4) + 2*(7C3*5C3) = 210 + 700 = 910".