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Given a standard $3 \times 3 \times 3$ rubik's cube, what are the total number of distinguishable configurations of the cube?

Two configurations are called distinguishable iff one can be rotated to make the other.

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The question specifies (except that it seems to be missing a negation) that states that differ only by rotating the entire cube in space do not count as distinguishable.

Additionally since the cube is "standard", states that differ only by 90° rotations of the center faces, without moving other visible parts of the cube, do not count as distinguishable. (This is not physically true if the stickers on these faces are not rotationally symmetric, which produces a "supercube").

With these assumptions, we can decide to count only states where the six center faces are in some standard positions, such as white center up, red center towards you (which then determines uniquely where the other four centers are).

  • There are 12 edge cubies, which can be assigned positions in the cube in $12!$ different ways.
  • Once we have decided on a position for each edge cubie there are two different orientations it can be put in. This gives an additional factor of $2^{12}$.
  • There are 8 corner cubies, which can be assigned positions in the cube in $8!$ different ways.
  • Once we have decided on a position for each corner, it can be rotated into three different orientations. This gives a factor of $3^8$.

Thus, the number of ways the cube can be re-assembled after taking it apart is $$ 12!\cdot 2^{12} \cdot 8! \cdot 3^{8} = 519,024,039,293,878,272,000 $$

Often one is only interested in the number of distinguishable states that are reachable from the solved state by legal moves. As described in this older question, exactly one in twelve states is reachable, so the number of reachable states is $$ \frac{12!\cdot 2^{12} \cdot 8! \cdot 3^{8}}{12} = 43,252,003,274,489,856,000$$

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Please refer to Wikipedia, there's detailed explanation.

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