  Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of
permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to
$m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim
\kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We
show that there exists a positive constant $C(m)$ such that, for all $n
\geqslant m$, \[C(m) \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1} \leqslant
\rho(n,m) \leqslant \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1}\] where
$\phi$ is Euler's totient function.
