group of 5 w/ at least 3 women formed from 12 women, 8 men

Out of 12 Women and 8 Men a group of 5 people is to be chosen. How many different groups can be formed if there must be at least 3 women in the group.

I have completed it as you could have 3W & 2M OR 4W & 1M OR 5W & 0M

12C3*8C2 + 12C4*8C1 + 12C5*8C0
= 10912

The query is why can we not:

12C3 * 17C2 based on the thinking that the 12C3 guarantees 3 women chosen and the 17C2 provides the choosing of 2 more people out of the 17 remaining. This answer is 29920. How is this thinking flawed?

Re: Combinations

With the second method there is no way to prevent an overcount.
You are counting the same groups of women more than once.