Symmetry Rule for Binomial Coefficients

Theorem

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

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

Proof
Follows directly from the definition.

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

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

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

Let $$0 \le k \le n$$.

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