Pascal's triangle

Saved part-way through to avoid any accidental loss of work. This article will hopefully be completed within the half hour.

Pascal's triangle is a triangular arrangement of the coefficients of a binomial expansion. It is constructed by placing a single 1, and then moving downward and out so that each number is the sum of the two numbers directly above it. (Elements outside the triangle are considered to be zero.)

The top row of the triangle (with just a 1) is the 0^th row, with rows below it 1^st, 2^nd, and so on. The left-most element in each row (which will always be a 1) is the 0^th element of that row, with the following elements continuing as 1^st, 2^nd, and so forth.

Applications

Pascal's triangle has many applications and properties. Some are useful, others are merely interesting. Only a few of the applications of Pascal's triangle are described below.