Introduction
Pascal's Triangle is named after mathematician Blaise
Pascalin. The rows of Pascal's triangle are conventionally enumerated starting with row 0, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. The procedure is shown in the below image.

PascalTriangleAnimated.gif
Pascal's triangle animated image

Application
Pascal's triangle determines the coefficients which arise in binomial expansion For an example, consider the expansion
(
x + y)2
=> x² + 2xy + y²

=>1y⁰ + 2 + 1x⁰.
Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like
x + y is raised to a positive integer power we have:
(
x + y)n = a0xn + a1xn−1y + a2xn−2y2 + … + an−1xyn−1 + anyn,
where the coefficients
ai in this expansion are precisely the numbers on row n of Pascal's triangle.