I ask this question mainly to resolve (hopefully) and error with the following problem.
The United States Court consists of $3$ women and $6$ men. In how many ways can a $3$-member committee be formed if each committee must have at least one woman?
My approach: Since each group needs at least one woman, there are $\binom31$ ways to pick the first one. After that, both men and women are allowed. There are $\binom82$ ways of doing this.
This amounts to $\binom31\times\binom82 = 84$ ways, but the answer is $64$. I don't see how this is the answer for two reasons. The first is that there should be at least somewhere a $3$ multiplied but $64$ has no powers of $3$. The second is that I wrote a counter in Java to look at all strings from the list $a,b,c,d,e,f,g,h,$ and $i$ choosen $3$ at a time and count how many words contained $a,b,$ or $c$.
What seems to be the problem?