Here is my approach so far:
Use a 10-gon as the representation of the necklace.
It follows that $10 \choose 3$ is the number of configuration of necklaces (3 being the number of white beads).
However, it is clear that we have overcounted (since we included rotations, e.g. 7 blacks followed by 3 whites and 3 whites followed by 7 blacks being counted separately). My approach is basically to look at the number of orbits in $D_5$.
|$D_5$| = 20; 1 + 9 rotations + 10 reflections.
My question is how I find |$G_x$|? I know $G_x$ is the stabilizer of a particular x, but I was wondering if I should find $G_x$ for a fixed x? What is X in this case? Do I just consider the stabilizers of each of the vertices of the 10-gon?
Thanks for your help.