Revision as of 21:27, 27 April 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: █

@@ Line 55: / Line 55: @@
 [https://www.youtube.com/watch?v=OMr9ZF1jgNc Pascal's Triangle and the Binomial Coefficients]<br />
 [https://www.youtube.com/watch?v=MVmgsATTg2I Example of choose function (Binomial Coefficient)]<br />
+[https://www.youtube.com/watch?v=lGow-vogneQ Binomial coefficients]<br />
 =References=
 [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 />
-[https://www.youtube.com/watch?v=lGow-vogneQ Binomial coefficients]<br />