Exercise 2, § 1

Let $m, n$ be integers such that $0 \leq n \leq m$. Let $p$ be a prime number and let $m = a_r + \alpha_1 p + \ldots + \alpha_n p'$ and $n = \beta_0 + \beta_1 p + \ldots + \beta_n p'$ be the expansions of $m$ and $n$ in base $p$, where we thus have $0 \leq \alpha_i < p, \ 0 \leq \beta_i < p$ for $0 \leq i \leq r$; suppose that $a_r \neq 0$. Show that $\binom{m}{n} \equiv 0 \pmod{p}$ if $\beta_i > a_r$ for at least one index $i$. If on the other hand, $\beta_i \leq \alpha_i$ for $0 \leq i \leq r$, then

(In $(\mathbf{Z}/p\mathbf{Z})[X]$ we have $(1 + X)^m = (1 + X)^{\alpha_0}(1 + X^p)^{\alpha_1} \ldots (1 + X^{p'})^{\alpha_r}$.)