there are 4 types of subcube:
those with no sides painted (1)
those with 1 sides painted (6)
those with 2 sides painted (12)
those with 3 sides painted (8)
Any shuffling of these subcubes within their own group will keep the cube exterior completely red, so there are 1!6!12!8! possible solutions. You should allow for orientations but i think this has no effect on this part of the calc (i think that for any given position, only 1 orientation is valid).
Step two: find the total number of outcomes (including orientations) and divide one by the other.
PS: I should warn you i dont have the best track record at combinatronics problems so i might have got that spectacularly wrong