Binomial Coefficient with Two

Theorem

 * $\forall r \in \R: \dbinom r 2 = \dfrac {r \paren {r - 1} } 2$

Corollary
The usual presentation of this result is:

Proof
From the definition of binomial coefficients:


 * $\dbinom r k = \dfrac {r^{\underline k}} {k!}$ for $k \ge 0$

where $r^{\underline k}$ is the falling factorial.

In turn:


 * $\ds x^{\underline k} := \prod_{j \mathop = 0}^{k - 1} \paren {x - j}$

When $k = 2$:
 * $\ds \prod_{j \mathop = 0}^1 \paren {x - j} = \frac {\paren {x - 0} \paren {x - 1} } {2!}$

where $2! = 1 \times 2 = 2$.

So:


 * $\forall r \in \R: \dbinom r 2 = \dfrac {r \paren {r - 1} } 2$

Also see

 * Particular Values of Binomial Coefficients for other similar results.