Given $K$ balls and $M$ buckets and the limits $L1$ and $L2$ where $0<L1<L2<M$. We distribute all balls in the buckets randomly so a bucket can end up with $0$ to $K$ balls.
How do I calculate the number of combinations that have a number of empty buckets $E$ where $L1<E<L2$?
As I understand it without the restrictions the number of combinations are $$ (K+M-1)!/(K!(M-1)!) $$ but I do not know where to go from here.
If it makes it easier we can add the restriction $K<=M$ or even $K=M$