Say that we have $k$ sets, each with cardinality $N$, where the elements in each set are taken at random from $M \ge N$ possible ones. The elements in each set are known to be distinct.

What is the probability that $j$ ($j=1,...,M$) elements appear at least once in the (union of the) sets?

We know that for $j = N$ the probability is $1$. For $j > N$ I wanted to use the binomial distribution, but the parameter $p$ should be conditioned somehow on the number of elements that already appear in the sets. Should I use some recursive formula for this?

**Edit**:

(1) This maybe also be formulated as the famous balls in bins problem, with the twist that the balls are thrown $N$ at a time to $M$ bins, where it is known that each group of $N$ balls falls into **distinct** $N$ bins.

(2) Another option to see this is as the (extended) Coupon Collector's Problem, who acquires $N$ (instead of $1$) distinct coupons at a time, from a set of $M$ distinct coupons. Denoting by $W\_t(j)$ the probability that he acquired $j$ distinct coupons at time $t$, my questions is what is the probability distribution of $W\_t(j)$, for $j=N,N+1,...,M$, for each time (or set in the original formulation) $t=1,2,...,k$. I saw something relevant here:
<https://math.stackexchange.com/questions/100175/probability-of-non-empty-bins-after-randomly-inserting-balls-by-pairs>