6 1600 Hint: Draw a Venn diagram, dividing $A \cup B \cup C$ into seven regions (as shown in the figure). First, arrange elements 1 and 2, with available regions being $p$ and $q$, each having 2 choices; then arrange elements 3 and 4, with available regions being $n$, $t$, $s$, and $r$, each having 4 choices; finally, arrange elements 5 and 6, with available regions being $m$, $n$, $t$, $s$, and $r$, each having 5 choices.
Therefore, the number of valid $(A, B, C)$ is
$$
2 \times 2 \times 4 \times 4 \times 5 \times 5=1600 \text{ (groups). }
$$
6 Let $A \cup B \cup C=\{1,2,3,4,5,6\}$, and $A \cap B=\{1,2\}$, $\{1$, $2,3,4\} \subseteq B \cup C$, then the number of $(A, B, C)$ that satisfy the conditions is $\qquad$. (Note: Different orders of $A$, $B$, $C$ are considered different groups)