Pascal's triangle

{\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\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}}

A diagram showing the first eight rows of Pascal's triangle.

In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients 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 mathematicians studied it centuries before him in Persia,^[1] India,^[2] China, Germany, and Italy.^[3]

The rows of Pascal's triangle 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. 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 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.

^ Cite error: The named reference :0 was invoked but never defined (see the help page).
^ Maurice Winternitz, History of Indian Literature, Vol. III
^ Peter Fox (1998). Cambridge University Library: the great collections. Cambridge University Press. p. 13. ISBN 978-0-521-62647-7.

[:0-1] Cite error: The named reference :0 was invoked but never defined (see the help page).

[2] Maurice Winternitz, History of Indian Literature, Vol. III

[roots-3] Peter Fox (1998). Cambridge University Library: the great collections. Cambridge University Press. p. 13. ISBN 978-0-521-62647-7.

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