Difference between revisions of "Binomial coefficient"

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The binomial coefficient ${n \choose k}$ are defined for positive integers $n$ and $k$ by the formula
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The binomial coefficient ${n \choose k}$ are defined for non-negative $n$ and $k$ by the formula
$${n \choose k} = \dfrac{n!}{(n-k)!k!}.$$
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$${n \choose k} = \dfrac{n!}{(n-k)!k!},$$
More generally, if $\alpha \in \mathbb{C}$ we define the binomial coefficient by  
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where $n!$ denotes the [[factorial]]. More generally, if $\alpha \in \mathbb{C}$ we define the (generalized) binomial coefficient by  
 
$${\alpha \choose k} = \dfrac{\alpha^{\underline{k}}}{k!},$$
 
$${\alpha \choose k} = \dfrac{\alpha^{\underline{k}}}{k!},$$
 
where $\alpha^{\underline{k}}$ denotes the [[falling factorial]].
 
where $\alpha^{\underline{k}}$ denotes the [[falling factorial]].
 
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
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File:Binomialcoefficient,n=20plot.png|Graph of $\displaystyle{20 \choose k}$.
 
File:Binomialcoefficientfunction.png|Graph of the complex binomial coefficient function.
 
File:Binomialcoefficientfunction.png|Graph of the complex binomial coefficient function.
 
</gallery>
 
</gallery>
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=Properties=
 
=Properties=
 
[[Binomial theorem]]<br />
 
[[Binomial theorem]]<br />
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[[Binomial series]]<br />
 
[[Binomial coefficient (n choose k) equals (n choose (n-k))]]<br />
 
[[Binomial coefficient (n choose k) equals (n choose (n-k))]]<br />
 
[[Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)]]<br />
 
[[Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)]]<br />
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=References=
 
=References=
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Binomial theorem|next=Binomial coefficient (n choose k) equals (n choose (n-k))}}: 3.1.2
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Binomial theorem|next=Binomial coefficient (n choose k) equals (n choose (n-k))}}: $3.1.2$
*[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_10.htm Abramowitz and Stegun]<br />
 
 
*[http://www.jstor.org/discover/10.2307/2975209?sid=21105065140641&uid=4&uid=70&uid=2&uid=3739256&uid=3739744&uid=2129 The Binomial Coefficient Function]<br />
 
*[http://www.jstor.org/discover/10.2307/2975209?sid=21105065140641&uid=4&uid=70&uid=2&uid=3739256&uid=3739744&uid=2129 The Binomial Coefficient Function]<br />
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 18:37, 25 September 2016

The binomial coefficient ${n \choose k}$ are defined for non-negative $n$ and $k$ by the formula $${n \choose k} = \dfrac{n!}{(n-k)!k!},$$ where $n!$ denotes the factorial. More generally, if $\alpha \in \mathbb{C}$ we define the (generalized) binomial coefficient by $${\alpha \choose k} = \dfrac{\alpha^{\underline{k}}}{k!},$$ where $\alpha^{\underline{k}}$ denotes the falling factorial.

Properties

Binomial theorem
Binomial series
Binomial coefficient (n choose k) equals (n choose (n-k))
Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)
Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))
Binomial coefficient (n choose 0) equals 1
Binomial coefficient (n choose n) equals 1
Sum over bottom of binomial coefficient with top fixed equals 2^n
Alternating sum over bottom of binomial coefficient with top fixed equals 0

Videos

Pascal's Triangle and the Binomial Coefficients
Example of choose function (Binomial Coefficient)
Binomial coefficients

References