There are $n$ balls to be distributed between $k$ bins. Balls of the same color are considered the same. How many distributions are there is there are $r$ red balls and $n-r$ blue balls?
I believe the problem should reduce to the number of ways of distributing $r$ red balls among $k$ bins, which is $r+k-1 \choose k-1$, multiplied by the number of ways of distributing $n-r$ blue balls among $k$ bins, which is $n-r+k-1 \choose k-1$.
Is this reduction justified?