 Solution

The binomial theorem isolates the only terms that can survive reduction modulo the prime number $p$:For $1\le r\le p-1$, neither $r!$ nor $(p-r)!$ is divisible by $p$, whereas $p!$ is. Consequently $p\mid \left(\genfrac{}{}{0ex}{}{p}{r}\right)$, and every intermediate term vanishes in modular arithmetic. ThusThis works for arbitrary integers $a,b$, including those divisible by $p$. It is the additive identity behind the Frobenius endomorphism in characteristic $p$.