Board Game Arena

Number of possible combinations in the game is 3.55361255095E+28.

I hope I calculated it correctly:
(56+12+12+1)! / (56!*12!*12!*1!)

Interesting fact

Looks alright, but can you prove that all of those are legally reachable positions? And how many are identical by symmetry/rotation?

Yes, they are all legally achievable.
But you are right about the symmetry. You could say that the positions are created symmetrically, so practically, i.e. from a game point of view, it is still divided by 4.

What's the actual measure you're calculating? combinations of what?
If you're looking at something like "winning" positions, then the total number of *valid* combinations realistically would ONLY be where 1 or more tiles exist on D5, F5, E4, or E6. There is no way to win the game without an adjacent tile

It's what's technically called "state space".