# Proof of the hockey stick/Zhu Shijie identity \sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}

$$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$

What’s the name of this identity? Is it the identity of the Pascal’s triangle modified.

How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?

EDIT 01 : This identity is known as the hockey-stick identity because, on Pascal’s triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.

Imagine the first $$n + 1n + 1$$ numbers, written in order on a piece of paper. The right hand side asks in how many ways you can pick $$k+1k+1$$ of them. In how many ways can you do this?
You first pick a highest number, which you circle. Call it $$ss$$. Next, you still have to pick $$kk$$ numbers, each less than $$ss$$, and there are $$\binom{s – 1}{k}\binom{s - 1}{k}$$ ways to do this.
Since $$ss$$ is ranging from $$11$$ to $$n+1n+1$$, $$t:= s-1t:= s-1$$ is ranging from $$00$$ to $$nn$$ as desired.