Binomial Coefficient of Prime/Proof 3

Proof
By the definition of binomial coefficient:

Now, $p$ divides the by Divisors of Factorial.

So $p$ must also divide the.

By hypothesis $k < p$.

We have that:


 * $k! = k \paren {k - 1} \dotsm \paren 2 \paren 1$

We also have that $p$ is prime, and each factor is less than $p$.

Thus $p$ is not a factor of $k!$.

Similarly:


 * $\paren {p - k}! = \paren {p - k} \paren {p - k - 1} \dotsm \paren 2 \paren 1$

It follows, by the same reasoning as above, that $p$ is also not a factor of $\paren {p - k}!$.

The result then follows from Euclid's Lemma.