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Difference between revisions of "Binomial Theorem"

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(Clarification. Seemed a bit incognoscible previously.)
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First invented by Newton, the Binomial Theorem states that for real or complex ''a'',''b'',<br><math>(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}</math> ''(The Binomial Theorem)''
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First invented by Newton, the Binomial Theorem states that for real or complex ''a'',''b'',<br><math>(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}</math>  
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We can understand why this is true, at least for integers, by following this simple observation:
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<math>\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math>
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This may be shown for the integers easily:<br>
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<br>Repeatedly using the distributive property, every way in which we can choose ''k'' ''a''s and ''n-k'' ''b''s is representing, leading to the formula.
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<center><math>\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math></center>
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<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 \binom{n}{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 07:19, 22 June 2006

First invented by Newton, the Binomial Theorem states that for real or complex a,b,
$(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}$

This may be shown for the integers easily:

$\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}$


Repeatedly using the distributive property, we see that for a term $\displaystyle a^m b^{n-m}$, we must choose $m$ of the $n$ terms to contribute an $a$ to the term, and then each of the other $n-m$ terms of the product must contribute a $b$. Thus the coefficient of $\displaystyle a^m b^{n-m}$ is $\displaystyle \binom{n}{m}$. Extending this to all possible values of $m$ from $0$ to $n$, we see that $(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}$.

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