Symmetry Rule for Binomial Coefficients/Proof 3

Theorem

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

Then:

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

Proof

If we choose $k$ objects from $n$, then we leave $n - k$ objects.

Hence for every choice of $k$, we are also making the same choice of $n - k$.

Hence the number of choices of $k$ objects is the same as the number of choices of $n - k$ objects.

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Two important relations