If $0 < i < p$, then $p \mid \binom{p}{i}$

Question

How do I prove this theorem?

For a prime number $p$ and integer $i$:

If $0 < i < p$ then $p \mid \binom{p}{i}$.

Answer 1

Hint: The simplest proof uses the relation $$\binom pk=\frac pk\binom{p-1}{k-1} \quad\text{for all}\; k>0 $$ and Gauss' lemma.