Binomial theorem

:For other topics using the name "binomial", see binomial (disambiguation).

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In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

Related Topics:
Mathematics - Formula - Power - Sum

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:(x+y)^n=sum_{k=0}^n{n choose k}x^ky^{n-k}quadquadquad(1)

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whenever n is any non-negative integer, the numbers

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:{n choose k}=rac{n!}{k!(n-k)!}

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are the binomial coefficients, and n! denotes the factorial of n.

Related Topics:
Binomial coefficient - Factorial

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This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to Chinese mathematician Yang Hui in the 13th century.

Related Topics:
Triangular arrangement - Blaise Pascal - 17th century - Yang Hui - 13th century

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For example, here are the cases n = 2, n = 3 and n = 4:

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:(x + y)^2 = x^2 + 2xy + y^2,

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:(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3,

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:(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.,

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Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a semiring as long as xy = yx.

Related Topics:
Real - Complex - Semiring

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