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Concept
Pascal's triangle
Summary
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients arising 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, India, China, Germany, and Italy.
The rows of Pascal's triangle are conventionally enumerated starting with row at the top (the 0th row). The entries in each row are numbered from the left beginning with 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.
The entry in the th row and th column of Pascal's triangle is denoted . For example, the unique nonzero entry in the topmost row is . With this notation, the construction of the previous paragraph may be written as follows:
for any non-negative integer and any integer . This recurrence for the binomial coefficients is known as Pascal's rule.
The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. The Persian mathematician Al-Karaji (953–1029) wrote a now-lost book which contained the first formulation of the binomial coefficients and the first description of Pascal's triangle. It was later repeated by Omar Khayyám (1048–1131), another Persian mathematician; thus the triangle is also referred to as the Khayyam's triangle (مثلث خیام) in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
Official source
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