Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 3 / 6D / d / Solution

A three-cycle is an even permutation, as is a product of two disjoint three-cycles. Thus every element counted in part (c) lies in $A_7$, and every order-three element of $A_7$ was already counted there. The answer remains $\boxed{350}$.