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The middle pieces of a rubik cube are fixed in place. They can't be moved so we can just ignore them. There are 12 middle side pieces and 8 corners pieces. So we would need to multiply all the different middle side piece possibilities with the corner piece possibilities and we should get the answer. But as you can't just swap one piece of a rubik cube, you need an even number of pieces swapped so we then need divide the answer by 2 to get the true answer.
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The 8 corner pieces each have 3 sides so the first corner we look at would have 8 x 3 different possibilities. The second corner would then have 7 x 3 different possibilities and so on and so forth. But for the last one, you don't need to calculate 3 × 1 as it's rotation is already set by the others. You can't just have one rotated corner on a rubik cube.
So the extended formel would thus be 8 × 3 × 7 × 3 × 6 × 3 × 5 × 3 × 4 × 3 × 3 × 3 × 2 × 3, or simplified: 8! × 3^7 just to calculate the corner possibilities.
To calculate the middle side possibilities you just need to repeat what did for the corner pieces and you'd end up with 12! x 2^11.
Now after combining those 2 formals and dividing the answer by two, as already explained, you would end with: 12! x 2^11 × 8! x 3^7 × 1/2 = 4.3252003e+19, which seems to confirm that the cube truly has an absurd number of possibilities.
And then there would also be possible combinations that would require you to disa.s.semble your cube and manually turn a piece. Then you would need to multiply the true answer with 6 (3 for the corner piece and 2 for the side piece) to get an ever larger amount of possibilities. And if you really want to mess things up, you could move the center pieces which would cause a whole lot of confusion and more possibilities by a factor of 6 x 5 x 4 x 3 x 2 or 6! or 720. So the new answer would be: 12! x 2^11 × 8! x 3^7 × 1/2 x 6 x 720 = 1.8684865e+23.
