Binomial Coefficient is Integer/Proof 1

Theorem

Let $\dbinom n k$ be a binomial coefficient.

Then $\dbinom n k$ is an integer.

Proof

If it is not the case that $0 \le k \le n$, then the result holds trivially.

So let $0 \le k \le n$.

By the definition of binomial coefficients:

\(\ds \binom n k\)

\(=\)

\(\ds \frac {n!} {k! \paren {n - k}!}\)

\(\ds \)

\(=\)

\(\ds \frac {n \paren {n - 1} \paren {n - 2} \cdots \paren {n - k + 1} } {k!}\)

The numerator is a product of $k$ successive integers.

From Factorial Divides Product of Successive Numbers, $k!$ divides it.

$\blacksquare$