  We study random generation in the symmetric group when cycle type
restrictions are imposed. Given $\pi, \pi' \in S_n$, we prove that $\pi$ and a
random conjugate of $\pi'$ are likely to generate at least $A_n$ provided only
that $\pi$ and $\pi'$ have not too many fixed points and not too many
$2$-cycles. As an application, we investigate the following question: For which
positive integers $m$ should we expect two random elements of order $m$ to
generate $A_n$? Among other things, we give a positive answer for any $m$
having any divisor $d$ in the range $3 \leq d \leq o(n^{1/2})$.
