Binomial Coefficient of Prime/Proof 1

Proof
Because:
 * $\dbinom p k = \dfrac {p \paren {p - 1} \paren {p - 2} \cdots \paren {p - k + 1} } {k!}$

is an integer, we have that:
 * $k! \divides p \paren {p - 1} \paren {p - 2} \cdots \paren {p - k + 1}$

But because $k < p$ it follows that:
 * $k! \mathrel \perp p$

that is, that:
 * $\gcd \set {k!, p} = 1$

So by Euclid's Lemma:
 * $k! \divides \paren {p - 1} \paren {p - 2} \cdots \paren {p - k + 1}$

Hence:
 * $\dbinom p k = p \dfrac {\paren {p - 1} \paren {p - 2} \cdots \paren {p - k + 1} } {k!}$

Hence the result.