Factors of Binomial Coefficient/Corollary 1

Theorem
For all $r \in \R, k \in \Z$:
 * $\paren {r - k} \dbinom r k = r \dbinom {r - 1} k$

from which:
 * $\dbinom r k = \dfrac r {r - k} \dbinom {r - 1} k$ (if $r \ne k$)

Proof
Then:
 * $\dbinom r k = \dfrac r {r - k} \dbinom {r - 1} k$

follows from the
 * $\paren {r - k} \dbinom r k = r \dbinom {r - 1} k$

by dividing both sides by $r - k$.