How many elements in $S_{8}$ are conjugate with $(12)(345)$? How many elements in $S_{8}$ are conjugate with $(12)(345)$?

My reasoning is as follows:
Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the number of elements of the type:
$(x_1 x_2)(x_3 x_4 x_5)$
There are $\frac{8\cdot7}{2}\cdot \frac{6 \cdot 5 \cdot 4}{3}=1120$ elements of this type. So $1120$ elements are conjugate with $(12)(345)$.
Is this correct?
5

Yes, your reasoning and answer are correct.
