binomial theoremBinomialTheorem2001-10-16 08:50:182005-02-22 07:54:13Theoremnumber theory combinatorics\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}The binomial theorem is a formula for the expansion of $(a+b)^n$, for $n$ a positive integer and $a$ and $b$ any two real (or complex) numbers, into a sum of powers of $a$ and $b$. More precisely,
$$(a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \cdots + b^n .
$$
For example, if $n$ is 3 or 4, we have:
\begin{eqnarray*}
(a+b)^3 &= a^3 + 3 a^2 b + 3 a b^2 + b^3 \\
(a+b)^4 &= a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 .
\end{eqnarray*}
This result actually holds more generally if $a$ and $b$ belong to a commutative \PMlinkname{rig}{Rig}.