On a Balanced Property of Compositions

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 $0leq rleq q$ 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 $lim_{n o infty} p_{n,r} = 1/q$. In fact, we show that an obvious necessary condition on $S$ turns out to be sufficient.