Balls into bins, probability that $k$ of the bins has at least 2 balls

Question

$n$ balls are thrown uniformly randomly into $m$ bins, I'd like to calculate the probability that exactly $k$ bins have at least 2 balls.

This is of course doable using the multinomial distribution, but the calculation becomes intractable for large cases - I'm still waiting on my Mathematica session to complete for a pretty small case.

Is there a more direct way of getting the result for this special case that is more efficient?

Answer 1

How about doing the old counting way? You can choose $k$ bins in $m \choose k$ ways and insert exactly two balls in each of them. You can choose these $2k$ balls in $n \choose {2k}$ ways. Each of the remaining $n-2k$ balls can distribute themselves in whatever way they like. This can be happen in ${(n-2k)}^m$ ways. The total possible distribution are $n^m$ ways. You can now get the probability by dividing these two.

Edit : So my answer does not account for the possibility of having more than $k$ bins when we let the $n-2k$ balls distribute themselves. Re-working my answer and will post an edit.