Difference between revisions of "Binomial coefficient"
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| − | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_10.htm Abramowitz and Stegun] | + | [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] | [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] | ||
Revision as of 01:53, 19 January 2015
The binomial coefficients are defined by the formula $${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$
Graph of the complex binomial coefficient function.
Properties
Proposition: $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$
Proof: █
Proposition: $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$
Proof: █
Proposition: ${n \choose 0} = {n \choose n} = 1$
Proof: █
Proposition: $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$
Proof: █
Proposition: $1 - \displaystyle {n \choose 1} + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$
Proof: █
Theorem (Binomial Theorem): $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$
Proof: █
