5. Cecilia has a six-sided die (numbered from 1 to 6) and 4 colors available. In how many ways can she color the six faces of the die using at least three different colors and ensuring that opposite faces are of different colors?
(A) $4^{3} \cdot 3^{3}$
(B) $3^{6}-2^{6}$
(C) $2^{6} \cdot 3^{2}$
(D) $2^{4} \cdot 3 \cdot 5 \cdot 7$
(E) None of the above
5. The answer is (D). First, let's count in how many ways we can color the cube so that opposite faces have different colors. We have $4 \cdot 3$ choices for faces 1 and 6, and the same for each of the pairs of faces 2 and 5 and 3 and 4. Overall, we have $4^{3} \cdot 3^{3}$ colorings. We must now subtract from them all the colorings that do not satisfy the first requirement, i.e., all those in which we have used only 2 colors. The number of such colorings is equal to the number of ways to choose 2 distinct colors from the 4 available, 6, multiplied by the number of possible colorings of the cube with those colors, $2^{3}$, since for each pair of faces we have 2 ways to use the two available colors. The number of admissible colorings is therefore $12^{3}-6 \cdot 8$, which is equal to $2^{4} \cdot 3 \cdot 5 \cdot 7$.