Twenty-seven identical white cubes are assembled into a single cube, and the outside of that cube is painted black.
The cube is then disassembled and rebuilt randomly.
What is the probability that the outside of this cube is again completely black?
There are 27! ways of placing the cubes in their places, times 24 ways of orientating them: T= 27! x 24.
For the random cube to be black:
The center all-white cube must go in the center, in any of 24 ways.
The 6 cubes with just one black face must go in the centers of the faces: 6! times 4 orientations.
The 12 cubes with two black faces must go in the middle of the edges: 12! times 2 orientations.
And the 8 cubes with three black faces must go in the corners: 8! times 3 orientations.
The total number of random all-black combinations is the product B= 24 x 6! x 4 x 12! x 2 x 8! x 3.
The chance that an all-black cube happens randomly is R/T= 2/(27 x 25 x 23 x 19 x 17 x 13 x 13 x 11 x 7), a very samll number indeed!