Difference between revisions of "Binomial coefficient"

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The binomial coefficients are defined by the formula
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The binomial coefficient ${n \choose k}$ are defined for non-negative $n$ and $k$ by the formula
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$${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$
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$${n \choose k} = \dfrac{n!}{(n-k)!k!},$$
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where $n!$ denotes the [[factorial]]. More generally, if $\alpha \in \mathbb{C}$ we define the (generalized) binomial coefficient by
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$${\alpha \choose k} = \dfrac{\alpha^{\underline{k}}}{k!},$$
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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.
 
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=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Binomial theorem]]<br />
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<strong>Proposition:</strong> $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$
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[[Binomial series]]<br />
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<div class="mw-collapsible-content">
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[[Binomial coefficient (n choose k) equals (n choose (n-k))]]<br />
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<strong>Proof:</strong> █
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[[Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)]]<br />
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</div>
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[[Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))]]<br />
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</div>
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[[Binomial coefficient (n choose 0) equals 1]]<br />
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[[Binomial coefficient (n choose n) equals 1]]<br />
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Sum over bottom of binomial coefficient with top fixed equals 2^n]]<br />
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<strong>Proposition:</strong> $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$
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[[Alternating sum over bottom of binomial coefficient with top fixed equals 0]]<br />
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<strong>Proof:</strong> █
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Proposition:</strong> ${n \choose 0} = {n \choose n} = 1$
 
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<strong>Proof:</strong> █
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Proposition:</strong> $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Proposition:</strong> $1 - \displaystyle {n \choose 1}  + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$
 
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<strong>Proof:</strong> █
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Theorem (Binomial Theorem):</strong> $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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=Videos=
 
=Videos=
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[https://www.youtube.com/watch?v=OMr9ZF1jgNc Pascal's Triangle and the Binomial Coefficients]
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[https://www.youtube.com/watch?v=OMr9ZF1jgNc Pascal's Triangle and the Binomial Coefficients]<br />
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[https://www.youtube.com/watch?v=MVmgsATTg2I Example of choose function (Binomial Coefficient)]
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[https://www.youtube.com/watch?v=MVmgsATTg2I Example of choose function (Binomial Coefficient)]<br />
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[https://www.youtube.com/watch?v=lGow-vogneQ Binomial coefficients]<br />
  
 
=References=
 
=References=
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[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_10.htm Abramowitz and Stegun]<br />
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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))}}: $3.1.2$
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[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]
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*[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 />
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[https://www.youtube.com/watch?v=lGow-vogneQ Binomial coefficients]
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[[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.

  • Graph of $\displaystyle{20 \choose k}$.

  • Graph of the complex binomial coefficient function.

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

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