Symmetry Rule for Binomial Coefficients/Proof 1

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Theorem

Let $n \in \Z_{>0}, k \in \Z$.

Then:

$\dbinom n k = \dbinom n {n - k}$


Proof

Follows directly from the definition of binomial coefficient, as follows.

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

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

In both cases:

$\dbinom n k = \dbinom n {n - k} = 0$


Let $0 \le k \le n$.

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

$\blacksquare$


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