# Pascals Triangle

Created | Updated Jan 28, 2002

In 1653, a french mathematician named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or equivalently the number of ways to choose

*n*objects from a group of

*m*indistinguishable objects. The numbers in each row also correspond to the coefficients of (1+

*x*)

^{n}

The first few rows of Pascals triangle look like this:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

In each row, the two outermost numbers are 1. All others are the sum of the two numbers above (e.g. in row 5, the 6 comes from 3+3).

It turns out that Pascal's triangle holds many interesting numeric patterns. One way of seeing some of these patterns is to pick a number

*x*and color all numbers in the triangle that are evenly divisible by

*x*with one color, and all the other numbers in the triangle with a second color. To see as much of the pattern as possible, you need to be able to see as many rows of the triangle as possible, but coloring a large number of rows like this by hand is very boring and time consuming. A computer can color many rows of the triangle in only a few seconds, so we can use it to look at the results using many different divisors.