Example 5 (2008 Chongqing College Entrance Examination Question) A person has 4 different colored light bulbs (with enough of each color), to be installed on the 6 points $A, B, C, A_{1}, B_{1}, C_{1}$ as shown in Figure 5, with the requirement that the light bulbs at the ends of the same line segment must be of different colors. The number of installation methods where each color of light bulb is used at least once is $\qquad$ (answer with a number).
Solution (Exclusion Method): Since points $A_{1}, B_{1}, C_{1}$ are pairwise of different colors, there are $\mathrm{A}_{4}^{3}$ ways to install the lights at points $A_{1}, B_{1}, C_{1}$. Select one position from $A$, $B$, $C$ to install the remaining color of the light bulb, which has $C_{3}^{1}$ installation methods. Without loss of generality, assume it is installed at $A$. At this point, $B$ can only install a light bulb of the same color as $A_{1}, C_{1}$, and $C$ can only install a light bulb of the same color as $A_{1}, B_{1}$. However, if both $B$ and $C$ install light bulbs of the same color as $A_{1}$, the condition is not satisfied. Therefore, the number of installation methods that meet the conditions is $3 \mathrm{~A}_{4}^{3} \times 2 \times 2 - 3 \mathrm{~A}_{4}^{3} = 216$.

Comment: This problem can also be solved using the method of discussing alternate positions.