Suppose I have $n$ sets of $k$ balls each, with each one of the $nk$ balls distributed uniformly at random among $m$ bins. Further suppose that I have a probability vector $p=(p_1,\dots,p_m)$. I am interested in selecting one ball from each of the $n$ sets, such that each bin $i$ contains roughly $p_i n$ selected balls. Is there an asymptotic expression for the number of ways this can be done, in the limit as $n\to\infty$?
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$\begingroup$ So you distribute the $nk$ balls and then try to select one per set to match $p$? $\endgroup$– Brian HopkinsCommented Jun 16, 2020 at 22:21
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$\begingroup$ @BrianHopkins yes, correct. $\endgroup$– Tom SolbergCommented Jun 16, 2020 at 22:22
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$\begingroup$ How do the other quantities depend on $n$? Are you keeping $k$, $m$, and $p$ constant as you take $n\to\infty$? $\endgroup$– Sam ZbarskyCommented Jun 17, 2020 at 6:16
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$\begingroup$ @SamZbarsky yes, everything else is fixed, and we can assume that $k\ll m$ if we want. $\endgroup$– Tom SolbergCommented Jun 17, 2020 at 15:13
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$\begingroup$ What "roughly" means? The answer may significantly depend on how one defines "roughly" here. $\endgroup$– Max AlekseyevCommented Jun 17, 2020 at 17:41
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