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Suppose we have $n$ bins and $k<n$ is some natural number. Each turn we select at random $k$ distinct bins out of the $n$ bins in which we place a ball. How many turns does it take on average until there is at least $1$ ball in each of the bins? This part has been solved in the comments (as it turned out to be a duplicate).

Related to this : If we call a collection of $k$ balls a winner if one of the balls is the first one to enter a bin, how many winners are there on average. So the first batch of ball has a 100$\%$ chance of being a winner and then the probability of being a winner decreases as more balls are added. For the coupon collector's problem mentioned in the comments (with $k=1$) it is obvious that there will always be exactly $n$ winning batches.

Darkwizie
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    When $k=1$ this is the coupon collector's problem, so you might start by researching that. – saulspatz Nov 13 '18 at 22:55
  • Thank you for this reference, it gives me a good idea of how difficult the problem is and a good starting point to investigate further! – Darkwizie Nov 13 '18 at 23:07
  • "Each turn we select at random k distinct bins out of the n bins in which we place a ball." Does it mean that $k $ balls are placed in (randomly chosen) $k $bins each turn? – user Nov 13 '18 at 23:14
  • @user : Yes exactly – Darkwizie Nov 13 '18 at 23:21
  • The first part is essentially the same as https://math.stackexchange.com/questions/2147576/probability-expected-number-of-draws-to-get-all-52-cards-at-least-once-drawing which has two good responses – Henry Nov 13 '18 at 23:42
  • Ok yes, I agree, we can therefore see the first part as solved. – Darkwizie Nov 14 '18 at 08:55

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