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 \( 0 \leq r < q \) be integers, with \( q \geq 2 \), and let \( p_{n,r} \) be the probability that \( m(a) \) is congruent to \( r \mod q \). 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.