Difference between revisions of "Binomial Theorem"
(Clarification. Seemed a bit incognoscible previously.) |
m (Fixed \binom{}{} to {} \choose {}) |
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This may be shown for the integers easily:<br> | This may be shown for the integers easily:<br> | ||
<center><math>\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math></center> | <center><math>\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math></center> | ||
− | <br>Repeatedly using the distributive property, we see that for a term <math>\displaystyle a^m b^{n-m}</math>, we must choose <math>m</math> of the <math>n</math> terms to contribute an <math>a</math> to the term, and then each of the other <math>n-m</math> terms of the product must contribute a <math>b</math>. Thus the coefficient of <math>\displaystyle a^m b^{n-m}</math> is <math>\displaystyle \ | + | <br>Repeatedly using the distributive property, we see that for a term <math>\displaystyle a^m b^{n-m}</math>, we must choose <math>m</math> of the <math>n</math> terms to contribute an <math>a</math> to the term, and then each of the other <math>n-m</math> terms of the product must contribute a <math>b</math>. Thus the coefficient of <math>\displaystyle a^m b^{n-m}</math> is <math>\displaystyle n \choose m</math>. Extending this to all possible values of <math>m</math> from <math>0</math> to <math>n</math>, we see that <math>(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}</math>. |
Revision as of 08:20, 22 June 2006
First invented by Newton, the Binomial Theorem states that for real or complex a,b,
This may be shown for the integers easily:
![$\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}$](http://latex.artofproblemsolving.com/a/9/d/a9de53f8bebe423d8614c4e58a9f1a0fb5244b30.png)
Repeatedly using the distributive property, we see that for a term , we must choose
of the
terms to contribute an
to the term, and then each of the other
terms of the product must contribute a
. Thus the coefficient of
is
. Extending this to all possible values of
from
to
, we see that
.