Pascal's triangle: Difference between revisions

Revision as of 16:40, 11 February 2025

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}}In [[mathematics]], '''Pascal's triangle''' is an infinite [[triangular array]] of the [[binomial coefficient]]s which play a crucial role in probability theory, [[combinatorics]], and algebra. In much of the [[Western world]], it is named after the French mathematician [[Blaise Pascal]], although other [[mathematician]]s studied it centuries before him in Persia,<ref name=":0" /> India,<ref>Maurice Winternitz, ''History of Indian Literature'', Vol. III</ref> China, Germany, and Italy.<ref name="roots">{{cite book |author=Peter Fox |title=Cambridge University Library: the great collections |url=https://books.google.com/books?id=xxlgKP5thL8C&pg=PA13 |year=1998 |publisher=Cambridge University Press |isbn=978-0-521-62647-7 |page=13}}</ref>

The rows of Pascal's triangle are conventionally enumerated starting with row <math>n = 0</math> at the top (the 0th row). The entries in each row are numbered from the left beginning with <math>k = 0</math> and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or anyand7 other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.

== Formula ==