Symmetry Rule for Binomial Coefficients

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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.

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 $0 \le k \le n$.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \binom n k$	$=$	$\displaystyle \frac {n!} {k! \ \left({n - k}\right)!}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {n!} {\left({n - k}\right)! \ k!}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {n!} {\left({n - k}\right)! \ \left ({n - \left({n - k}\right)}\right)!}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \binom n {n - k}$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \binom n k\)	\(=\)	\(\displaystyle \frac {n!} {k! \ \left({n - k}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {n!} {\left({n - k}\right)! \ k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {n!} {\left({n - k}\right)! \ \left ({n - \left({n - k}\right)}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \binom n {n - k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)