Vandermonde's identity states that for all positive integers

,

,

, the following identity holds [1]:

.

In the special case of

, the identity reduces to

.

It is a well-known fact of Pascal's triangle that the

entry of the

row is

(where indexing for rows and entries starts from 0), and the number at the bottom of the diamond is

. Thus, it is sufficient to show that the sum of the squares of the elements in the middle row is equal to the number at the bottom of the diamond.

It is also well-known that the number of paths from the top element to another element, traveling only in the steps down left or down right, is equal to the value of the element. Thus, the number of red paths from the top of the diamond to the bottom is

, and the number of paths from the top element to entry

in the middle row is

. By symmetry, there are also

paths from row

, entry

to the bottom element of the diamond, so there are

paths from the top element of the diamond to the bottom passing through element

in the middle row. Each path from the top element to the bottom element must pass through exactly one element in the middle row, so that

.

Snapshot 1: Pascal's triangle for 18 rows; there are actually 19 rows, since row indices start at 0

Snapshot 2: Pascal's triangle for four rows and a path passing through the 2 in the middle row

Snapshot 3: Pascal's triangle for 14 rows and a path passing through a 35 in the middle row

This was a project for Advanced Topics in Mathematics II, 2017–2018, Torrey Pines High School, San Diego, CA.