My question is a variant of the balls-in-bins problem (similar to the coupon collector problem), where I'm interested in how many of the $n$ bins will be empty after throwing $k$ balls.

As far as I understand, this normally equals $\mathbf{E}[X] = n\left(\frac{n-1}{n}\right)^k$. Where I get lost is when I try and extend this problem as follows:

Rather than $n$ bins that can hold any number of balls, let us say that $m$ of the bins are no longer available after receiving one ball (i.e. without replacement). The other $n-m$ bins continue to act as normal. If $m=n$, then of course $X=n-k$ bins will be empty after throwing $k$ balls.

For other values of $m$ though, how does this affect the number of empty bins?

Let me know if I need to clarify anything. This is my first time posting! Thank you for your help :)


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