Combinations question - why is my approach wrong? I'm learning Permutations and Combinations and while trying to solve this simple question, I stuck on the solution:-
From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
The solution was to consider all three possibilities one by one- 3 men and 2 women, 4 men and 1 women, all 5 men. Calculate the combinations for each case and add them.
But I was trying it from a different approach- Firstly select 3 men from 7 (gives 35 combinations) and multiply it with the combinations of selecting 2 more committee members from 10 remaining members. So my answer came out to be 35 x 45 = 1575. But the answer is 756 which is not even a multiple of 35. So is my approach wrong? Why so?

If you first select 3 men and then fill up with two arbitrary people, then you count each combination with exactly three men once, but others are counted repeatedly. For example, you count all-male committees 10 times, once for each way to choose the three first men in retrospect from the final five men.
Smaller numbers example: Given John, Jack and Jill, form a committee of two people with the constraint that the committee must have at least one man. Actually, there are three such committees possible (because the constraint is no constraint at all).
But your method would give four: First select John as male, then select either Jack or Jill as second. Or first select Jack as male, then select John or Jill as second. The committee "Jack and John" is counted twice by you.
