If we have a deck of 48 different cards and 4 players each get 12 cards, it is well known how to calculate the number of possible distributions: take fac(48) and divide 4 times by fac(12).

In a german card came (Doppelkopf) there are 24 different card types, but 2 copies of each type, that is 48 cards at all.

How many distributions when there are 4 players? My first approach was to assume there are 48 different cards, and then to divide by 2 and by 2 (... 24 times). But this underestimates the real number of distributions. Because: Say player 1 has both spade kings. This has been ruled out already by dividing by fak(12). But if player 1 and player 2 both have a spade king, then we must divide by 2.

How to calculate the number of different distibutions?

Is there a closed formula as in the single card deck case?

If we have a deck of $48$ different cards and $4$ players each get $12$ cards, it is well known how to calculate the number of possible distributions: $\frac{48!}{12!12!12!12!}$

In a german card came (Doppelkopf) there are $24$ different card types, but $2$ copies of each type, that is $48$ cards at all.

How many distributions when there are $4$ players? My first approach was to assume there are $48$ different cards, and then to divide by $2^{24}$. But this underestimates the real number of distributions. Because: Say player 1 has both spade kings. This has been ruled out already by dividing by $12!$. But if player 1 and player 2 both have a spade king, then we must divide by 2.

How to calculate the number of different distibutions?

Is there a closed formula as in the single card deck case?