Difference between revisions of "Binomial coefficient"

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=References=
 
=References=
 
[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]
 +
[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 22:39, 13 January 2015

The binomial coefficients are defined by the formula $${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$


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:

References

Abramowitz and Stegun The Binomial Coefficient Function