The question is the distribution of the number of empty cells.
Imagine to have $N$ cells, and I have $k$ balls to distribute randomly (uniform) among $N$ cells so that no more than 1 ball can occupy the cell. where $k\leq N$.
Then we repeat the same procedure $n$-times.
Now the other balls of different "types" can go in the cells where other-type of balls are.
After the complete process:
- What is the probability that a randomly chosen cell is empty?
- The expected number of empty cell?
- What changes if I remove the condition of only one ball per "type"?
___________ Sketch of solution _________
the probability that each of n specified cells will be occupied is ${{N}\choose{k}}^{-1}$, and there are ${{N}\choose{k}}^n$ random allocations of $kn$ balls among the $N$ cells.