Example 7.11 Find the number of first-class circular permutations formed by selecting 6 elements from the 3-element set $A=\{a, b, c\}$ with repetition allowed.
The number of the first type of circular permutations is
$$
\begin{aligned}
T_{3}(6) & =\frac{1}{6} \sum_{d \mid 6} \varphi(d) \cdot 3^{\frac{6}{d}} \\
& =\frac{1}{6}\left[\varphi(1) \cdot 3^{6}+\varphi(2) \cdot 3^{3}+\varphi(3) \cdot 3^{2}+\varphi(6) \cdot 3\right] \\
& =\frac{1}{6}\left(3^{6}+3^{3}+2 \cdot 3^{2}+2 \cdot 3\right)=130 .
\end{aligned}
$$