How to prove this equation of combinations?

Question

$$\sum_{i=k}^n\binom ik=\binom{n+1}{k+1}$$ $$n,k\in N^+;n\ge k$$

Or it can be expressed as:

$$\sum_{i=k}^ni(i-1)\cdots(i-k+1)=\frac{n+1}{k+1}n(n-1)\cdots(n-k+1)$$ $$n,k\in N^+;n\ge k$$

Answer 1

This comes by induction from $$\binom{n+1}{k+1}-\binom{n}{k+1}=\binom{n}k.$$

Answer 2

Suppose that you have $n+1$ cards, numbered $1,2,\dots,n+1$ and you want to draw $k+1$ card out of them. The number of possibilities is $\displaystyle \binom{n+1}{k+1}$.

If the largest number on the drawn card is $k+1$, then the other $k$ drawn cards should be from those cards numbered $1,2,\dots,k$. Number of possibilities is $\binom{k}{k}$.

If the largest number on the drawn card is $k+2$, then the other $k$ drawn cards should be from those cards numbered $1,2,\dots,k+1$. Number of possibilities is $\binom{k+1}{k}$.

In general, if the largest number on the drawn card is $i+1$ (where $i\ge k$), then the other $k$ drawn cards should be from those cards numbered $1,2,\dots,i$. Number of possibilities is $\binom{i}{k}$.

So the total number of possibilities is $\displaystyle \sum_{i=k}^n\binom{i}{k}$.

Therefore, $\displaystyle \sum_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$.