I'm trying to find the number of elements conjugate to (12)(34)(56789) in S12 but don't know how to go about it.
Any help is much appreciated.
First we form a 5-cycle, there are 12 elements and we only choose 5 of them therefore there are $\displaystyle {{12}\choose 5}$ ways to choose elements for a 5-cycle. Once we have chosen these 5 elements there are $\displaystyle 4!$ ways of ordering them to form distinct 5-cycles. Now we need to form a 2-cycle, there are 7 elements left and we choose 2, therefore, $\displaystyle {7\choose 2}$ and once those have been chosen we have $\displaystyle {5\choose 2}$ for the remaining 2-cycle. Therefore, we get $\displaystyle \frac{1}{2}\cdot 4! {{12}\choose 5}{7\choose 2}{5\choose 2}$ because we overcounted the 2-cycles twice i.e. $\displaystyle (12345)(67)(89)$ is the same as $\displaystyle (12345)(89)(67)$.