How many sets of four vertices of a cube are coplanar?
I think twelve. The six faces, or course, to start with.
For the other six: Consider looking at one face of the cube. Visualize a diagonal of that face, then push that line through the cube to the opposite face. There's one plane through four vertices. You can do that for each of the two diagonals on each face. This might lead one to believe that there would be twelve of them, but you have to remember that the one you push through from this side is the same as one pushed through from the other side, so you have to halve the number.