Suppose n balls are distributed in m urns. The probability that the first r urns receive k balls is
$$\frac{\binom{n}{k}r^k(m-r)^{n-k}}{m^n}$$
I am most confused about the $r^k$ part. I know there is some relation to the binomial distributions, but I am having trouble visualizing what a "success" here is?
There are $m^n$ ways the $n$ balls can be placed in the $m$ urns.
If the first $r$ urns get $k$ balls, then the remaining $m-r$ urns will get the remaining $n-k$ balls. There are $r^k$ ways to place $k$ balls in $r$ urns, and $(m-r)^{n-k}$ ways to place $n-k$ balls in $m-r$ urns. Finally, there are ${n\choose k}$ ways to choose the $k$ balls out of the $n$. Multiplying these give $${n\choose k}r^k(m-r)^{n-k}$$ and dividing by the number of possible choices $$\frac{{n\choose k}r^k(m-r)^{n-k}}{m^n}$$ gives the desired probability.