  We give a short proof for an explicit upper bound on the proportion of
permutations of a given prime order $p$, acting on a finite set of given size
$n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv
k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot
k!)^{-1}$ with equality if and only if $p\leq n<2n$.
