This excerpt from Herstein's book. As you see here the author writes the formula $o(G)=\sum \limits c_a$. And he writes that index $a$ is taken from each conjugate class. I guess that $a$ should be taken from distinct conjugate classes. What if $C(a)=C(b)$ for $a\neq b$ then $c_a=c_b$ and we count this $c_a$ with factor $2$.
Am I right that index $a$ should be taken from distinct conjugate classes?
The conjugacy classes are automatically distinct. Assume they are not distinct. Then there is a $b\in G$ such that $b\in C(a)\cap C(a')$, so $b\sim a$ and $b\sim a'$, hence $a\sim a'$ and $C(a)=C(a')$. So either they are disjoint, or they coincide.