HMMT 2014 C4

Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq\{1,2,3, \ldots, 8\}$. (b) $|A \cap B|=|B \cap C|=|C \cap A|=2$. (c) $|A|=|B|=|C|=4$.

Here, $|S|$ denotes the number of elements in the set $S$.