How many different circular necklaces containing ten beads can be made using beads of at most two colors?
I know I need calculate situation with 2 colors, 8 second color, 3 first color, 7 second color, 4 first color, 6 second color and 5 first color, 5 second color.
But I have problems with my calculations:
$$\frac{1}{20}\left(\frac{10!}{8!2!} + 3\times 5 + 1\times 5 + 3\times 15\right) = \frac{110}{20}. $$
Please show to me how to solve this problem correctly.