: Mathematics: binomial theorem

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binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial $(x + y) n$ into a sum involving terms of the form $ax b y c$ , where the exponents $b$ and $c$ are nonnegative integers with $b + c = n$ , and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$ . For example, for $n = 4$ , ${\\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}$ The coefficient $a$ in the term of $ax b y c$ is known as the binomial coefficient ${\\displaystyle {\ binom {n}{b}}}$ or ${\\displaystyle {\ binom {n}{c}}}$ (the two have the same value). These coefficients for varying $n$ and $b$ can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where ${\\displaystyle {\ binom {n}{b}}}$ gives the number of different combinations of $b$ elements that can be chosen from an $n$ -element set. Therefore ${\\displaystyle {\ binom {n}{b}}}$ is often pronounced as " $n$ choose $b$ ".
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Binomial_theorem)

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