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 a 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 kth entry in the nth row of Pascal's triangle is the binomial coefficient "n choose k", written ${\tbinom {n}{k}}$ . The rows are enumerated from $n=0$ at the top (the 0th row), and the entries in each row are numbered from $k=0$ on the left to $k=n$ on the right, and are usually staggered left relative to the numbers in the previous row. The triangle may be constructed as follows: the top entry is equal to 1, and each lower entry is the sum of the two entries above it to the left and 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 first two numbers 1 and 3 in row 3 are added to produce the number 4 below them 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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