the theorem giving the expansion of a binomial raised to any power.
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- binomial coefficient,
- binomial distribution,
- binomial experiment,
- binomial nomenclature,
- binomial series,
Origin of binomial theorem
First recorded in 1865–70
Dictionary.com Unabridged Based on the Random House Unabridged Dictionary, © Random House, Inc. 2019
a mathematical theorem that gives the expansion of any binomial raised to a positive integral power, n . It contains n + 1 terms: (x + a) n = x n + nx n – 1 a + [ n (n –1)/2] x n – ² a ² +…+ (n k) x n – k a k + … + a n, where (n k) = n !/(n–k)! k !, the number of combinations of k items selected from n
Collins English Dictionary - Complete & Unabridged 2012 Digital Edition © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012
The theorem that specifies the expansion of any power of a binomial, that is, (a + b)m. According to the binomial theorem, the first term of the expansion is xm, the second term is mxm-1y, and for each additional term the power of x decreases by 1 while the power of y increases by 1, until the last term ym is reached. The coefficient of xm-r is m![r!(m-r)!]. Thus the expansion of (a + b)3 is a3 + 3a2b + 3ab2 + b3.
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