How do you calculate the number of permutations of a Rubik's Cube depending on the amount of pieces?

Answer 1

For a 3x3x3 Rubik's cube,

there are 8 corners, each with 3 sides

each of the 8 corners can have 8permutations(placements/positions), and therefore it is 8!(eight factorial, or 8^8)

each of the 8 corners can have 3 orientations, and therefore it is 3^8

There are 12 edges, each with 2 sides.

Each of the 12 edges can have 12 permutations, and therefore it is 12!

Each of the edges can have 2 orientations, and that means that it is 2^12

the equation is

8!*3^8*12!*2^12

=519 024 039 293 878 272 000

(519 quintillion if you're too lazy to read it)

but only 1/12 combinations are possible without taking apart the cube

so

519 024 039 293 878 272 000/12

=43 252 003 274 489 856 000

OR

8! corner permutations

3^8 orientations, but the last corner's orientation depends on the other 7's, so it is really 3^7

12! edge permutations, but since there is an even permutation of corners, the edges are even, too so 12!/2

2^12 edge orientations, but the last edge depends on the preceding edge's orientation, so 2^11

8!*3^7*12!/2*2^11=43 252 003 274 489 856 000