So, there is this Italian card game called "Scopa" (broom) I used to play when I was a kid (still do sometimes), and I've been wondering how many possible games can be played.

I'll try to explain how the game is structured but not the exact rules since they're not important for the calculation (if you're interestend in the game here's the [Wikipedia article](https://en.wikipedia.org/wiki/Scopa)).

The deck is made of $40$ cards and the game is played in 2. At the beginning, of the 40 cards, 4 are dealt on ground, and then 3 cards are dealt to each player.
At this point the players will each play one card at a time, until the cards in their hands end, and then they will be dealt 3 cards each again. This is repeat till the deck is empty.

What I've been thinking is:

1. The number of ways the initial 4 cards can be dealt is $\binom{40}{4}$
2. The number of ways the 3 cards can be dealt to the players is $\binom{36}{3}$ the first time cards are dealt, $\binom{33}{3}$ the second time, and so on...
3. Each time a player receives the 3 cards, he has $3!$ ways of playing those cards.

Put all together, my estimates gives: $$\binom{40}{4}\cdot3!\cdot\prod\_{i=1}^{12}\binom{i\cdot3}{3}\simeq9.37\cdot10^{37}$$

Is my reasoning correct?