23. Use 4 red beads and 3 blue beads to make a necklace. How many different necklaces can be made?
23. $D=\{1,2,3,4,5,6,7\} \quad R=\{r, b\}, C$ has $2^{\top}$ coloring methods,
$$
\begin{array}{l}
G=\left\{e, \rho_{7}, \rho_{7}^{2}, \rho_{7}^{3}, \rho_{7}^{4}, \rho_{7}^{5}, \rho_{7}^{6}, \tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}, \tau_{5}, \tau_{6}, \tau_{7}\right\}, \\
P_{G}\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}\right)=\frac{1}{14}\left(x_{1}^{7}+6 x_{7}+7 x_{1} x_{2}^{3}\right), \\
P_{G}\left((r+b),\left(r^{2}+b^{2}\right),\left(r^{3}+b^{3}\right), \cdots,\left(r^{7}+b^{7}\right)\right) \\
\quad=\frac{1}{14}\left[(r+b)^{7}+6\left(r^{7}+b^{7}\right)+7(r+b)\left(r^{2}+b^{2}\right)^{3}\right],
\end{array}
$$

The coefficient of $r^{4} b^{3}$ is $\frac{1}{14}\left(\mathrm{C}_{7}^{4}+7 \mathrm{C}_{3}^{2}\right)=\frac{1}{14}(35+21)=4$.