Show How to Prove $\binom{n}{r}$ with Pascal's Triangle

[JGalway]

Repeatedly apply $\binom{n}{r}= \binom{n-1}{r}+\binom{n-1}{r-1}$ to show:

$$\binom{n}{r}=\sum_{i=1}^{r+1}\binom{n-i}{r-i+1}$$

The closest i got was showing you could show different iterations with the binomial coefficients (Pascal's Triangle).

 

[Euge]

Hi JGalway,

To prove the result formally, I suggest using the principle of mathematical induction.

 

[JGalway]

Hi JGalway,

To prove the result formally, I suggest using the principle of mathematical induction.

Thanks for the reply.
If it's just induction I think I will just ignore it I thought i was missing some property of ${n \choose r}$.(Never really liked induction)