Counting & Probability · Superchallenge problem

How many ordered triples $(A,B,C)$ of subsets of $\{1,2,3,4,5,6\}$ satisfy all of the following: $A\cup B\cup C=\{1,2,3,4,5,6\}$; $A\cap B\cap C=\varnothing$; and each of $A\cap B$, $B\cap C$, $C\cap A$ is nonempty?