The formula answers: how many tuples $(\sigma_1,\sigma_2,\dots,\sigma_n)$ of elements of a given group $G$ such that (1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class. (2) $\sigma_1\sigma_2...\sigma_n= \text{id}$.
I want to know the name and exact content of this formula. Also, is there any connection between this formula and Burnside counting theorem (orbit-counting theorem)? I also want of a proof of the formula using idempotent (perhaps) and other related theorems.
Thanks in advance.