Two versions of class equation

Question

I have seen two versions of class equation.

1. If $\bar{x}_1$, ..., $\bar{x}_m$ where $x \in G$ are distinct conjugacy classes of $G$, then $|G|= \displaystyle\sum_{n=1}^{m} [G:C_G(x_i)]$.

2. Sometimes it can be written as $|G|= |C(G)|+\displaystyle\sum_{n=1}^{m} [G:C_G(x_i)]$.

I understand that these two are basically the same, but any subtle difference here?

Thanks in advance!

Answer 1

As @AndreadCaranti points out in his comment . . .

These two results are the same, given that, in the second, the $x_i$ ranges over all non-central classes.

Answer 2

If $m$ is the number of all the conjugacy classes, then the second one is wrong, as it counts twice the singletons.