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.
An image is supposed to go here. You can help us out by creating one and. Thanks.
This identity can be proven by induction on .
Base case Let .
Inductive step Suppose, for some , . Then .
It can also be proven algebraicly with pascal's identity
Look at It can be rewritten as Using pascals identity, we get We can continuously apply pascals identity until we get to
Vandermonde's Identity states that , which can be proven combinatorially by noting that any combination of objects from a group of objects must have some objects from group and the remaining from group .