Pascal's Triangle

Pascal's triangle is a triangular array of counting numbers. It begins with a first line of 1. The elements of subsequent rows are found by adding the two elements diagonally above in the preceding row (e.g., 10 is the sum of the 4 and 6 above it). The outermost elements of each row are always 1, because there is no second number above to contribute to the sum.

					1
				1		1
			1		2		1
		1		3		3		1
	1		4		6		4		1
1		5		10		10		5		1

There are many ways to derive the values in the triangle. See Ask Dr. Math: FAQ: Pascal's Triangle for a discussion of the triangle and several methods for generating it. See the links from that page to learn more about Pascal's Triangle. Also look at Ladja's Report on Pascal's Triangle and Algebra Through Problem Solving, Chapter 1: The Pascal Triangle.

For help generating pictures of the remainder patterns see:

Coloring Remainders in Pascal's Triangle Once at the site, click on "increase depth" at the bottom of the diagram to see additional rows. Click on the "What?" and "How?" buttons for instructions and additional resources.
Pascal's Triangle Interface, Pascal's Triangle, and Pascal's Triangle.
Cut the Knot: Dot Patterns and the Sierpinski Gasket places the patterns that emerge in Pascal's Triangle in the context of fractal geometry.