Probability of complete coupon set in $K$ boxes where a box has $N$ distinct coupons?

Say there is a contest to collect a full set of $M$ coupons.

Each box of product has $N$ distinct coupons of the $M$ possible coupons in it, selected uniformly without replacement from the $M$ possible coupons, the same $N$ in number for all boxes.

I'm interested in the probability that after opening $K$ boxes of product I have at least one complete set of the $M$ coupons.

My attempt has been to take the probability of a coupon not being in a box, use that to determine the probability of that coupon not being in any of the $K$ boxes, and then... I get flummoxed, and don't think this is the correct approach.

I searched the site for coupon collector problems, there are a couple that deal with similar situations (multiple in a box), but with the box having coupons selected with replacement, so duplication in a box is possible.

Right now I'm using a CAS to get probability of sums of hypergeometric samples having all bins covered to check my ideas, but that's utterly impractical for other than tiny cases.

Is this not as trivial as I think it is, or is there a direct way to calculate this?

Solution 1:

The probability can be obtained by inclusion-exclusion. (The probability in the standard coupon collector's problem is given by a Stirling number of the second kind, which can likewise be calculated using inclusion-exclusion.)

There are $\binom Mj$ ways to select $j$ coupons that weren't obtained, and the probability not to obtain these $j$ coupons in $K$ boxes is $\left(\frac{\binom{M-j}N}{\binom MN}\right)^K$, so the probability to obtain all $M$ coupons is

$$ {\binom MN}^{-K}\sum_{j=0}^M(-1)^j\binom Mj\binom{M-j}N^K\;. $$