Question 86
For a triplet $(a, b, c)$ of positive integers, an $a \times b \times c$ rectangular parallelepiped is constructed by assembling $abc$ unit cubes. Then, all the faces of the parallelepiped are colored. How many possible triplets $(a, b, c)$ are there if the number of colored unit cubes equals the number of uncolored ones? Regard two triplets to be the same if they are permutations of each other.