View source for Pascal's triangle

In [[mathematics]], '''Pascal's triangle''' is a triangular array of the [[Binomial coefficient|binomial coefficients]].

== Description ==

[[File:320px-Pascal's triangle 5.svg.png|thumb]]
In the Western world, it is named after French mathematician [[Blaise Pascal]], although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.

The rows of Pascal's triangle (sequence A007318 in OEIS) are conventionally enumerated starting with row n = 0 at the top (the 0th row).

The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

Having the indices of both rows and columns start at zero makes it possible to state that the binomial coefficient \tbinom{n}{k} appears in the nth row and kth column of Pascal's triangle.

== Animated example ==

In Pascal's triangle, each number is the sum of the two directly above it.

[[File:PascalTriangleAnimated2.gif|thumb]]

This animation shows this relation in the construction of the first five rows, however the pattern applies for an infinite range.

This version has the 1 cells already filled in, and includes actual animation to better demonstrate the construction.

== See also ==

* [[Bean machine]], [[Francis Galton]]'s "quincunx"
* [[Bell triangle]]
* [[Binomial coefficient]]
* [[Binomial expansion]]
* [[Euler triangle]]
* [[Fibonacci sequence]]
* [[Floyd's triangle]]
* [[Leibniz harmonic triangle]]
* [[Maestro (game)]]
* [[Mathematical beauty]]
* [[Mathematics]]
* [[Mathematics and art]]
* [[Multiplicities of entries in Pascal's triangle]] (Singmaster's conjecture)
* [[Pascal matrix]]
* [[Pascal's pyramid]]
* [[Pascal's simplex]]
* [[Proton NMR]], one application of Pascal's triangle
* [[(2,1)-Pascal triangle]]
* [[Star of David theorem]]
* [[Trinomial expansion]]
* [[Trinomial triangle]]

== External links ==

* [https://en.wikipedia.org/wiki/Pascal%27s_triangle#See_also Pascal's triangle] @ Wikipedia
* [http://www.cut-the-knot.org/arithmetic/combinatorics/PascalTriangleProperties.shtml Patterns in Pascal's Triangle] @ cut-the-knot.org

=== Pascal's triangle and pinball ===

* [http://commons.bcit.ca/math/entertainment/pascaltr/ The Pinball Game] @ bcit.ca - "Pascal's triangle is very useful for analysing the pinball machine"
* [http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/binomials/MyPascal/lo/51_pinballprob/51_pinballprob_00.htm Binomial probability – Pinball] @ nsw.edu.au

=== YouTube videos ===

* [https://www.youtube.com/watch?v=tQshSPTTpfg Pascal's 3D Triangle]

=== Play My Code ===

* [http://www.playmycode.com/play/game/victorbs/pascals-triangle Pascal's Triangle] by victorbs



[[Category:Mathematics]]