Example: Sets $A, B, C$ (not necessarily distinct) satisfy $A \cup B \cup C=\{1,2,3, \cdots, 11\}$. Find the number of solutions of the ordered triples $(A, B, C)$ that meet the condition.
As shown in Figure 1-1, the intersection of sets $A, B, C$ results in 7 regions, so each element of $1,2,3, \cdots, 11$ has 7 choices for its position. Therefore, the number of solutions for the ordered triplet $(A, B, C)$ is $7^{11}$.