Coupon collectors problems with multiple coupons in a cereal box

In the coupon collectors problem, there are $n$ unique coupons and each cereal box has 1 coupon only. I would like to modify the problem such that there are $m$ boxes of cereal in total and each box has $c_i (1 \le c_i \le n)$ number of coupons.

Then how many boxes of cereal do I need to buy to have $n$ unique coupons?

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Since the coupons within a box are not guaranteed to be different, basically we are just buying boxes in batches of $c_i$.

As in the original coupon problem, we expect to have to buy $n \log n$ original boxes to make for a complete set of $n$ unique coupons. So if each new box contains $c$ coupons, we expect to need to buy $\dfrac{n \log n}c$.

However, without any specification on how the $c_i$ are distributed for the new boxes we cannot make an estimate of the number of boxes we need to buy in the new situation.

Let us tackle the general situation where each box has the same probability distribution, say chance $p_i$ to contain $i$ boxes, $1 \le i \le n$. Then each box is expected to contain $$E = \sum_{i=1}^n ip_i$$ coupons. It follows that we expect to have to buy $$\frac{n \log n}E$$ boxes to complete a set.