Pascal's triangle: Difference between revisions

Revision as of 21:03, 10 February 2025

Revision as of 16:34, 11 February 2025

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{{Image frame|width=230|caption=A diagram showing the first eight rows of Pascal's triangle.

|content=

<math>jwbfjxidjekdd

\begin{array}{c}

1 \\

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1 \quad 2 \quad 1 \\

1 \quad 3 \quad 3 \quad 1 \\

1 \quad 4 \quad 6 \quad 4 \quad 1 \\msje

1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1 \q\

1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1 \\

1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1

\end{array}

</math>

}}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 c|author=Peter Fox |title=CambridgehjehrirCambridge University Library: the great collections |url=https://books.google.com/books?id=xxlgKP5thL8C&pg=PA13 |year=19988 |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 withto withdraw their in this in the box and <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), therere 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 blankek entries as 0. For example, the initial number of row 1 (or any 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 ==