The number of elements of order 3 in $S_7$ is?

Question

The number of elements of order 3 in $S_7$ is? In $S_7$ there are 2 different disjoint cycles of order 3 i.e (3_)(3_)(1_) and (3_)(1_)(1_)(1_)(1_) but how to calculate no. of elements of order 3 .

Answer 1

There are $\binom{7}{3}\cdot2=70$ three-cycle elements. There are $\binom{7}{3}\cdot2\cdot\binom{4}{3}\cdot2\cdot\frac{1}{2}=280$ elements built of two disjoint three-cycles. Unless I am missing something that would cover all elements of order $3$, so there are $350$ of them.