How many elements of S8 have the same cycle structure as (12)(345)(678)? Should I do this by counting all the numbers using a factorial and then multiply by 8? Can any one show me the specific steps?
 A: HINT:


*

*First choose a transposition; in how many ways can you do that?  

*Let $k$ be the smallest of the $6$ remaining elements. Choose $2$ other elements to go in the same $3$-cycle as $k$. In how many ways can you do that? The remaining $3$ elements will of course form the other $3$-cycle.  

*Finally, how many different $3$-cycles can you make from a given set of $3$ elements?  


Put the pieces together, and you’ll have your answer.
A: The number of ways to choose $2$ out of $8$ is $\dbinom 8 2 = 28$.
Having done that, the number of ways to choose one of the $3$-cycles from the remaining $6$ elements is $\dbinom 6 3 = 20$, but that counts everything twice, because choosing and choosing the complementary $3$-cycle are really both the same permutation.
Finally, there is the choice between these:
$$
\begin{array}{ccccc}
& & 1 \\
& \swarrow & & \nwarrow \\
2 & & \longrightarrow & & 3
\end{array}
\qquad\text{ and }\qquad
\begin{array}{ccccc}
& & 1 \\
& \nearrow & & \searrow \\
2 & & \longleftarrow & & 3
\end{array}
$$
That choice must be made in each $3$-cycle.
So $28\times\dfrac{20}2\times 2\times 2$ is what you're looking for. 
