Multinomial coefficients modulo a prime

Question

Let $p$ be a prime and let $m \geq 1$. Lucas' theorem implies that the binomial coefficient ${p^m-1 \choose k}$ is not divisible by $p$ for any $0 \leq k \leq p^m-1$. I wonder if something similar holds for multinomial coefficients ${p^m-1 \choose k_1, \ldots, k_s}$ where $k_1, \ldots, k_s$ are non-negative integers such that $k_1 + \cdots + k_s = p^m-1$. That is, what are sufficient and necessary conditions for such multinomial coefficients to not be divisible by $p$? Thanks!