My problem is the following: throw (randomly, independently) $p$ balls in $n$ bins. What is the expected number of balls in the $k$ emptiest bins?


I have some results about the expected number of bins with $x$ balls (that would be $n {p \choose x} (\frac{1}{n})^x (1-\frac{1}{n})^{p-x}$) but I can't deduce what I want from that... 


Did I miss something obvious? Any pointers?



expected number of balls in k emptiest bins
The most interesting/relevant thing i found was in the Newman-Shepp's generalization of the coupon collector problem, which seems to be the exact dual problem of the balls in the emptiest bin ( = "how many balls do you have to throw to ensure there are $x$ in every bin [and thus in the emptiest] ")


According to <http://en.wikipedia.org/wiki/Coupon_collector%27s_problem#Extensions_and_generalizations>


...the expected number is $ p = n\log n + (x-1)n\log\log n + O(n)$


So my guess for the emptiest bin would be to invert this formula (express $x$ in function of the rest) , and i have a bound on the number in k emptiest, quite good if $ k << n$ (well, the estimation being for $n \to \infty$, this seems okay :-) )


Maybe i should see the proofs of this to see if there are ideas that can be adapted & formalized. 


