Factors of Binomial Coefficient

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

where $\displaystyle \binom r k$ is a binomial coefficient.

Hence:
 * $\displaystyle \binom r k = \frac r k \binom {r - 1} {k - 1}$ (if $k \ne 0$)

and:
 * $\displaystyle \frac 1 r \binom r k = \frac 1 k \binom {r - 1} {k - 1}$ (if $k \ne 0$ and $r \ne 0$)

Also, for all $r \in \R, k \in \Z$:
 * $\displaystyle \left ({r - k}\right) \binom r k = r \binom {r - 1} k$

from which:
 * $\displaystyle \binom r k = \frac r {r - k} \binom {r - 1} k$ (if $r \ne k$)

Proof
If $k = 0$ then $\displaystyle k \binom r k = r \binom {r - 1} {k - 1} = 0$ by definition.

Otherwise:

If $k \ne 0$, we can divide both sides of:
 * $\displaystyle k \binom r k = r \binom {r - 1} {k - 1}$

by $k$ to obtain:
 * $\displaystyle \binom r k = \frac r k \binom {r - 1} {k - 1}$

If $k \ne 0$ and $r \ne 0$, we can divide both sides of:
 * $\displaystyle \binom r k = \frac r k \binom {r - 1} {k - 1}$

by $r$ to obtain:
 * $\displaystyle \frac 1 r \binom r k = \frac 1 k \binom {r - 1} {k - 1}$

Finally:
 * $\displaystyle \binom r k = \frac r {r - k} \binom {r - 1} k$

follows from the
 * $\displaystyle \left ({r - k}\right) \binom r k = r \binom {r - 1} k$

by dividing both sides by $r - k$.