Corollary 1 (binomial formula)

Tag 05H4, permanent. Cite it as [Bourbaki, Tag 05H4].

Algebra, chapter I, § 8. Rings

Let x and y be two elements of a commutative ring A. Then:

(x + y)^n = \sum_{p=0}^n \binom{n}{p} x^p y^{n-p}.

Formula (14) applied to $L = \{1, 2\}$ , $x_1 = x$ and $x_2 = y$ gives

(x + y)^n = \sum_{p+q=n} \frac{n!}{p! q!} x^p y^q,

the sum extending over ordered pairs of positive integers $p, q$ with $p + q = n$ . The binomial formula follows immediately from this (Set Theory, III, § 5, no. 8).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 2, § 8
Proposition 9, § 1

Cites

*Set Theory*, III, § 5, no. 8