finding the summation of $\sum_{\ell=k}^n \binom{\ell}{k}$

Question

$$\sum_{\ell=k}^n \binom{\ell}{k} = \binom{n+1}{k+1}$$

Can someone explain how we got that equation? Expanding it we get

$$\binom{k}{k} + \binom{k+1}{k} + \binom{k+2}{k} + .....+ \binom{n}{k}$$

but how does that add up to $$\binom{n+1}{k+1}$$

Yes it's a duplicate, but I had no idea it was called the Hockey-Stick Identity. — Rockstar5645, Jun 02 '17 at 16:36
Then you should've closed it as a duplicate yourself rather than waiting for us to do it for you. — Simply Beautiful Art, Jun 02 '17 at 16:41
While a duplicate is pending, it should appear at the top of your question "This may answer your question... " and you'll have the option of "yes, this answers my question" or "no, my question is different". — Simply Beautiful Art, Jun 02 '17 at 16:46

score 2 · Accepted Answer · answered Jun 02 '17 at 16:36

2

Hint : Rewrite $$\binom{k}{k} =\binom{k+1}{k+1}$$

And now use the identity $$\binom{k}{r}+\binom{k}{r+1}=\binom{k+1}{r+1}$$

answered Jun 02 '17 at 16:36

Jaideep Khare

1 Answers1