Solution
= Solution
An element of order three in $S_7$ has <cycle type> $3\,1^4$ or $3^2 1$. There are
$$
\binom73(3-1)!=35\cdot2=70
$$
single three-cycles. For two disjoint three-cycles, choose the two unordered three-element supports and an orientation on each:
$$
\frac12\binom73\binom43(2)(2)=280.
$$
Hence $S_7$ contains $\boxed{350}$ elements of order three.