We worked on two different cube coloring problems before. One is to paint a unit cube with $1$ face in red, $1$ face in green, $1$ face in yellow, and $3$ faces in blue color. The other is to paint a unit cube with $2$ faces each in red, green and blue.
By using Pólya enumeration theorem or Bunside’s lemma, the number of ways with $m$ colors to paint a cube can be calculated with the following formula: $$c(m)=\dfrac{1}{24}(m^6+3m^4+12m^3+8m^2)$$ The above formula includes coloring with $1$ color, $2$ colors, …, $m$ colors. For example, $$c(2)=10\ \ \ \ \ \ \ \ \ c(3)=57$$ The ways to pain the cube with 6 different colors and each face having a different color is $\dfrac{6!}{24}=30$ because the Rotational Symmetries of the Cube is 24.