1. Can someone show me a combinatorial proof of the hockey stick identity? http://stuff.daniel15.com/cgi-bin/ma...+1%7D%7Br+1%7D. I've done it for a specific case say when http://stuff.daniel15.com/cgi-bin/mathtex.cgi?f%20=%205 and http://stuff.daniel15.com/cgi-bin/mathtex.cgi?r%20=%202. But I don't know how to generalise it.
2. How many different ordered triples http://stuff.daniel15.com/cgi-bin/ma...gi?%28a,b,c%29 of non-negative integers are there such that http://stuff.daniel15.com/cgi-bin/mathtex.cgi?a+b+c=50?
How would you do this question using partitions?
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?