The distribution of the Carlitz binomial coefficients modulo a prime

For a nonnegative integer $n$, and a prime $\mathrm{\wp }$ in ${\mathbb{F}}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0\le m\le n$ for which the Carlitz binomial coefficients ${\left(\genfrac{}{}{0px}{}{n}{m}\right)}_C$ fall into each of the residue classes modulo $\mathrm{\wp }$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.