Consider a group of 4 boys and 6 girls.
In how many ways can one choose a subset of 2 boys and 3 girls?
I know there is a quicker way of doing this than manually but i don't know the method.
Do you know the combination formula? Like in the binomial formula?
$\displaystyle \begin{pmatrix}n \\k \end{pmatrix} = \frac{n!}{(n-k)!k!}$
That gives you the number of ways to choose k things from a set of n elements when order does not matter. Clearly in a subset, the order of choosing the things doesn't make a difference. Do you think you can take it from here?
Yup looks good to me. That's what I got when I did it.
When you have n ways to do one thing, and m ways to do another, and they are independent of eachother (like in this situation, which boys you pick has no influence on what girls you pick) the total number of ways to do it is $\displaystyle n\cdot m$. That is why you multiply them instead of add them.