How many elements are in the conjugacy class of $\tau \in S_9$? Just one simple question:
Let $\tau =(56789)(3456)(234)(12)$.
How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?
First step is to write it in disjunct cyclces I guess. What's next? :)

This is an exercise from BAI (Jacobson) that yields to a combinatorial argument

Let the partition [of the integer $n$] associated with a conjugacy class be $(n_1,n_2,\dots,n_q)$ where $$n_1=\cdots=n_{q_1}>n_{q_1+1}=\cdots=n_{q_1+q_2}>n_{q_1+q_2+1}\cdots$$
Show that the number of elements in this conjugacy class is $$\frac{n!}{\prod q_i!\prod n_j}$$

