I might not have made my question clear, but I wanted to derive the distribution from a purely counting point of view.

So, what I wanted to ask actually was if there is a way to prove \(\displaystyle \frac{\binom{np}{k}\binom{nq}{r-k}}{n^r} = \binom{r}{k}p^{k}q^{r-k} \), which is where I got stuck.