A population of $n$ elements includes $np$ red ones and $nq$ black ones ($p+q = 1$). A random sample of size $r$ is taken with replacement. Show that the probability of its including exactly $k$ red elements is $${r \choose k} p^k q^{r-k}.$$
Attempt: the probability should be $\frac{{np \choose k}{nq \choose r-k}}{{n \choose r}}$. By manipulating this, I get ${r \choose k} \frac{n-r \choose np-k}{n \choose np}$. So, the problem is reduced to finding the equality: $p^k q^{r-k} = \frac{n-r \choose np-k}{n \choose np}$. But, I am stuck here. How can I prove this equality?