Symmetry Rule for Binomial Coefficients/Proof 1

Theorem

 * $\displaystyle \forall n \in \Z, n > 0: \forall k \in \Z: \binom n k = \binom n {n - k}$

where $\displaystyle \binom n k$ is a binomial coefficient.

Proof
Follows directly from the definition, as follows.

If $k < 0$ then $n - k > n$.

Similarly, if $k > n$, then $n - k > 0$.

In both cases $\displaystyle \binom n k = \binom n {n - k} = 0$.

Let $0 \le k \le n$.