Pascal’s triangle

Pascal’s triangleMathworldPlanetmath, also called Tartaglia’s triangle, is the following configurationMathworldPlanetmathPlanetmath of numbers:


In general, this triangle is constructed such that entries on the left side and right side are 1, and every entry inside the triangle is obtained by adding the two entries immediately above it. For instance, on the fourth row 4=1+3.

Historically, the application of this triangle has been to give the coefficients when expanding binomial expressions. For instance, to expand (a+b)4, one simply look up the coefficients on the fourth row, and write


Pascal’s triangle is named after the French mathematician Blaise Pascal (1623-1662) [3]. However, this triangle was known at least around 1100 AD in China; five centuries before Pascal [1]. In modern languagePlanetmathPlanetmath, the expansion of the binomial is given by the binomial theoremMathworldPlanetmath discovered by Isaac Newton in 1665 [2]: For any n=1,2, and real numbers a,b, we have

(a+b)n = k=0n(nk)an-kbk
= an+(n1)an-1b+(n2)an-2b2++bn.

Thus, in Pascal’s triangle, the entries on the nth row are given by the binomial coefficientsMathworldPlanetmath


for k=1,,n.

Pascal’s triangle has many interesting numerical properties. For example, it is easy to see that the sum of the entries in the nth row is 2n. This can be easily proved by inductionMathworldPlanetmath, but a more elegant proof goes as follows:


If you look at the long diagonals parallel to the diagonal sides of the triangle, you see in the second diagonal the integers 1,2,3,4,. The next diagonal down contains the triangular numbersMathworldPlanetmath 1,3,6,10,15,, and the row below that the tetrahedral numberMathworldPlanetmath 1,4,10,20,35,. It is easy to see why this is: for example, each triangular number is the sum of the previous triangular number and the next integer, which precisely reflects the arrangement of the triangle. Each tetrahedral number is the sum of the previous tetrahedral number and the size of the next “layer” of the tetrahedron, which is just the next triangular number. Similarly, succeeding diagonals give “triangular” number in higher dimensions.


  • 1 Wikipedia’s on the binomial coefficients
  • 2 Wikipedia’s on Isaac Newton
  • 3 Wikipedia’s on Blaise Pascal
Title Pascal’s triangle
Canonical name PascalsTriangle
Date of creation 2013-03-22 13:36:56
Last modified on 2013-03-22 13:36:56
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Topic
Classification msc 05A10
Synonym Tartaglia’s triangle
Related topic BinomialCoefficient
Related topic PascalsRule