  Let $S$ be a finite set of positive integers with largest element $m$. Let us
randomly select a composition $a$ of the integer $n$ with parts in $S$, and let
$m(a)$ be the multiplicity of $m$ as a part of $a$. Let $0\leq r<q$ be
integers, with $q\geq 2$, and let $p_{n,r}$ be the probability that $m(a)$ is
congruent to $r$ modulo $q$. We show that if $S$ satisfies a certain simple
condition, then $\lim_{n\to \infty} p_{n,r} =1/q$. In fact, we show that an
obvious necessary condition on $S$ turns out to be sufficient.
