Binomial theorem

Statement

Let $R$ be a commutative ring, let $x, y \in R$ , and let $n$ be a natural number. Then $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}.$ Here $\binom{n}{k}$ is the binomial coefficient, and the integer $\binom{n}{k}$ acts on $R$ by repeated addition.

Remarks

For $n = 2$ it gives $(x + y)^2 = x^2 + 2xy + y^2$ . For $n = 3$ it gives $x^3 + 3x^2 y + 3x y^2 + y^3$ . The formula stays true in a noncommutative ring when $xy = yx$ . With $x = y = 1$ it gives the sum of binomial coefficients. With $x = -1$ and $y = 1$ it gives the alternating sum. The multinomial theorem extends it to sums of more than two terms.

Reading path

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Binomial coefficient definition
Binomial theorem statement

Tag	`0EHWPWGHB2`
URI	`m2://0EHWPWGHB2`
Kind	statement, theorem
Writing	W1 stated
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 0EHWPWGHB2`.
Uses	Binomial coefficient
Same as	Wikidata Q26708 Wikipedia Binomial theorem

Binomial theorem

Statement

Remarks

Reading path

Further reading

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