Difference between revisions of "Binomial coefficient"
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| − | The binomial | + | 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]]. | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| + | 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 series]]<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+1) choose k) equals (n choose k) + (n choose (k-1))]]<br /> | |
| − | + | [[Binomial coefficient (n choose 0) equals 1]]<br /> | |
| − | + | [[Binomial coefficient (n choose n) equals 1]]<br /> | |
| − | + | [[Sum over bottom of binomial coefficient with top fixed equals 2^n]]<br /> | |
| − | + | [[Alternating sum over bottom of binomial coefficient with top fixed equals 0]]<br /> | |
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=Videos= | =Videos= | ||
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=References= | =References= | ||
| − | * {{BookReference|Handbook of mathematical functions | + | * {{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]<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.
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $3.1.2$
- The Binomial Coefficient Function
