Example 24 (2002 Wuhan City Some Schools College Entrance Examination Simulation Question) Let $I=\{1,2,3,4,5,6\}$, $A$ and $B$ are subsets of $I$. If $A \cap B=\{1,3,5\}$, then $(A, B)$ is called an "ideal pair of sets". The number of all "ideal pairs of sets" is ( ).
A. 9
B. 6
C. 27
D. 8
Solution: Choose C. Reason: We can establish the following mathematical model: As shown in Figure 4-5, let $A \cap\left(\complement_{1} B\right)$ be mailbox (1), $\left(\complement_{1} A\right) \cap B$ be mailbox (2), and $\complement_{1}(A \cup B)$ be mailbox (3). Elements 1, 3, 5 are in $(A \cap B)$, and elements 2, 4, 6 are considered as three letters. Thus, the problem is transformed into “how many ways are there to put three letters into three mailboxes.” By the multiplication principle, there are $C_{3}^{1} \times C_{3}^{1} \times C_{3}^{1}=27$ ways, meaning the total number of “ideal pairings” is 27.