1. ## Combinatorics

1. Can someone show me a combinatorial proof of the hockey stick identity? $\binom{r}{r}+\binom{r+1}{r}+\binom{r+2}{r}+ \cdots + \binom{f}{r} = \binom{f+1}{r+1}$. I've done it for a specific case say when $f = 5$ and $r = 2$. But I don't know how to generalise it.

2. How many different ordered triples $(a,b,c)$ of non-negative integers are there such that $a+b+c=50$?

How would you do this question using partitions?

2. ## Hockey stick identity

Assume f>r. The RHS can be understood as the number of ways to choose r+1 items from f+1 items. Imagine f+1 green balls lined up in front of you. Imagine that the r+1 items will be colored orange. There must be a rightmost orange ball. Suppose it's in the f+1 position. The other r balls must be chosen from f green balls. Now suppose the rightmost orange ball is in the f position. The other r balls must be chosen from f-1 positions. Can you see that these are the LHS terms, from rightmost to leftmost? Do you see why the sum ends at r choose r?