Factors of Binomial Coefficient

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

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

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

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

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

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

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

Otherwise:

$$ $$ $$ $$ $$ $$

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

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

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

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

$$ $$ $$ $$

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

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

by dividing both sides by $$r - k$$.