So I came across with the following exercise:
How many elements are conjugated with $a=(3\ 5\ 1\ 6)$ in $S_8$?
The solution was:
we are looking for $|Stab(a)|=|\{b\in S_8 : bab^{-1} = a\}|$ meaning $|Stab(a)|=\frac{|S_8|}{|S_8\cdot a|}$ when $|S_8 \cdot a|$ is the size of the conjugacy classes of $a$, meaning $|S_8 \cdot a|=\binom{8}{4}\frac{4!}{4}$. So we get: $|Stab(a)|=\frac{8!}{\binom{8}{4}\frac{4!}{4}}=96$.
I do understand why $|S_8|=8!$ but why $|S_8 \cdot a|=\binom{8}{4}\frac{4!}{4}$?