How many possible combinations? 4 x 4 grid of tiles (with caveats)

Question

I'm designing a board game, the surface of which is made up of 16 tiles, in a grid of 4x4 with 10 unique designs. 6 pairs of these are twins. The user can rotate and also rearrange the tiles and make their own configurations. I got curious about the number of possible permutations. NOTE that 2 of the 16 tiles have a circle design and so have a single state (looks the same when rotated), and one pair has a design that has only 2 states (90 degrees and 180 degrees look exactly like 270 and 360 degrees respectively). All tiles can appear anywhere in the grid. Any chance an equation exists for this or any way to calculate the number of possible permutations? Thanks!

Answer 1

How many ways can we place down $16$ unique square tiles onto this grid, ignoring rotations? There are $16! = 20,922,789,888,000$ ways to do this. If we account for repetition of the tiles, there are $6$ pairs of $2$, which can be freely swapped, so $16!$ over-counts by a factor of $2!^6 = 64$. So, still ignoring rotations, there are $$\frac{16!}{2!^6} = 326,918,592,000$$ possible placements into this grid.

Now, taking into account rotations, $12$ of the tiles have no symmetry whatsoever, and can be rotated independently. Then, two more tiles have $180^\circ$ rotational symmetry. This will add in a factor of $4^{12}\cdot 2^2 = 67,108,864$, This gives us the final count of: $$\frac{16!}{2!^6} \cdot 4^{12}\cdot 2^2 = 16! \cdot 4^{10} = 21,939,135,329,599,488,000,$$ or just shy of $22$ quintillion possibilities.