Symmetry Rule for Binomial Coefficients/Proof 1

Theorem

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

Then:

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

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 $

$=$

$\ds \binom n {n - k}$

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Two important relations
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text{B}$