Suppose we have a card game with some $n$ cards and $m$ players, where $m \mid n$. Each player starts with $\frac{n}{m}$ cards.
How many starting states does the game have?
How many states can the game be in after $k$ rounds, if in every round, every player drops $j$ of their remaining cards at random?
If $l = \frac{n}{m}$ , we have
$$\frac{n!}{(l!)^m}$$
starting states.
After $k$ rounds, every player has $l-kj$ cards left.
So, the total number of cards is $m(l-kj) = n-mkj$
So, the number of states is now
$$\binom{n}{n-mkj}\frac{(n-mkj)!}{(l-kj)!^m}$$