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'binomial theorem'
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| Title of object: |
binomial theorem |
| Canonical Name: |
BinomialTheorem |
| Type: |
Theorem |
| Created on: |
2001-10-16 08:50:18 |
| Modified on: |
2003-11-10 14:22:20 |
| Classification: |
msc:11B65 |
| Keywords: |
number theory combinatorics |
Revision comment (for changes between this and next version):
| Changes for correction #3218 ('This entry is broken in page images mode.'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
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*} |
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