Hockey-stick identity


In combinatorial mathematics, the identity
is known as the hockey-stick or Christmas stocking identity. That name stems from the graphical representation of the identity on Pascal's triangle: when the addends represented in the summation and the sum itself are highlighted, the shape revealed is vaguely reminiscent of those objects.

Proofs

The inductive and algebraic proofs both make use of Pascal's identity:

Inductive proof

This identity can be proven by mathematical induction on.
Base case
Let ;
Inductive step
Suppose, for some,
Then

Algebraic proof

We use a telescoping argument to simplify the computation of the sum:

A [combinatorial proof]

Imagine that we are distributing indistinguishable candies to distinguishable children. By a direct application of the stars and bars method, there are
ways to do this. Alternatively, we can first give candies to the oldest child so that we are essentially giving candies to kids and again, with stars and bars and double counting, we have
which simplifies to the desired result by taking and, and noticing that :

Another combinatorial proof

We can form a committee of size from a group of people in
ways. Now we hand out the numbers to of the people. We can divide this into disjoint cases. In general, in case,, person is on the committee and persons are not on the committee. This can be done in
ways. Now we can sum the values of these disjoint cases, getting