# Proving a binomial identity

• Apr 16th 2009, 03:07 PM
chaotixmonjuish
Proving a binomial identity
Prove that:

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

Here is as far as I've gotten:

$\displaystyle r\frac{(r-1)!}{(r-1-k)!(k)!}$

which would be equal to
$\displaystyle r\binom{r-1}{r-k}$...i think
I'm stuck for the next step.
• Apr 16th 2009, 03:46 PM
TheEmptySet
Quote:

Originally Posted by chaotixmonjuish
Prove that:

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

Here is as far as I've gotten:

$\displaystyle r\frac{(r-1)!}{(r-1-k)!(k)!}$

which would be equal to
$\displaystyle r\binom{r-1}{r-k}$...i think
I'm stuck for the next step.

So

$\displaystyle \binom{r-1}{k}=\frac{(r-1)!}{k!(r-1-k)!}$

Now if we multiply by

$\displaystyle \frac{(r-1)!}{k!(r-1-k)!}\cdot \frac{r}{(r-k)}=\frac{r!}{k!(r-k)!}=\binom{r}{k}$
• Apr 16th 2009, 03:56 PM
chaotixmonjuish
Well I'm stuck as to what to factor out to get the $\displaystyle \frac{r}{r-k}$. Im not too certain at all as to how the denomenator r-k comes into play.