I want to figure out how many elements of $S_5$ with the same cycle type as (12)(345).
I think the way to go about doing this would be to say :
There are $5C3=10$ ways to choose the elements in the 3-cycle and then $2C2=1$ way to choose the elements in the transposition.
As permutations are in general non-commuative there are also 2 ways to order the transposition and the 3-cycle (i.e. the transposition first or the 3 cycle first )
So in total there are $10 \times 1 \times 2=20 $ elements of this type.
Is this reasoning correct ?