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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.
and the binomial coefficient is the coefficient of the x2 term. Arranging the numbers in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle, satisfying the recurrence relation. The binomial coefficients occur in many areas of mathematics, and especially in combinatorics.
Hockey-stick identity. Recurrence relations of binomial coefficients in Pascal's triangle. Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n =6, r =2: 1+3+6+10+15=35. In combinatorial mathematics, the hockey-stick identity, [1] Christmas stocking identity, [2] boomerang identity, Fermat's identity or ...
Pascal's pyramid's first five layers. Each face (orange grid) is Pascal's triangle. Arrows show derivation of two example terms. In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. [1]
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .
Trinomial triangle. The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's triangle) entries above it: The -th entry of the -th row is denoted by. Rows are counted starting from 0.
In 1665 Pascal posthumously published his results on the eponymous Pascal's triangle, an important combinatorial concept. He referred to the triangle in his work Traité du triangle arithmétique (Traits of the Arithmetic Triangle) as the "arithmetic triangle". [4] In 1662, the book La Logique ou l’Art de Penser was published anonymously in ...
Generic Pascal's m -simplex. [edit] Let m (m > 0) be a number of terms of a polynomial and n (n ≥ 0) be a power the polynomial is raised to. Let m denote a Pascal's m - simplex. Each Pascal's m - simplex is a semi-infinite object, which consists of an infinite series of its components. Let mn denote its n th component, itself a ...