How many containers contain at least one item?

Question

I have X containers with Y slots each. The containers are not unique.

I also have n items that are to be placed in any of the X * Y slots at random. n is always less than X * Y.

How would I find the number of containers with at least one item that is statistically most likely? Is this possible to determine theoretically?

Answer 1

I suspect this may be quite difficult in general, though for $Y=1$ the answer is $n$

But a related question is the expected number of containers with exactly one item (i.e. the mean rather than the mode): that would be $X\dfrac{{Y \choose 1}{XY-Y \choose n-1}}{XY \choose n} = \dfrac{n(XY-Y)!(XY-n)! }{(XY-Y-n+1)!(XY-1)!}$

You then edited the question to at least one item. Here the expected number is $X\left(1-\dfrac{XY-Y \choose n}{XY \choose n}\right)$