Example 7.13 Find the number of first-class circular permutations made from 2 $a$s, 2 $b$s, 2 $c$s.
The number of the first type of circular permutations sought is
$$
\begin{aligned}
T(2,2,2) & =\frac{1}{6} \sum_{d \mid 2} \varphi(d) \frac{\left(\frac{6}{d}\right)!}{\left(\frac{2}{d}\right)!\left(\frac{2}{d}\right)!\left(\frac{2}{d}\right)!} \\
& =\frac{1}{6}\left[\varphi(1) \frac{6!}{2!\cdot 2!\cdot 2!}+\varphi(2) \frac{3!}{1!\cdot 1!\cdot 1!}\right] \\
& =\frac{1}{6}(90+6)=16 .
\end{aligned}
$$