Binomial Identity

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/BinomialIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BinomialIdentity.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identities

(∑_(k=0)^n)(({{n}, {k}})) (x^k) (a^(n-k))==(((a+x))^n)

Simplify[Sum[Binomial[n,k](x^k)(a^(n-k)),{k,0,n}],a>0]

(x+y) (((-a n+x+y))^(n-1))==(∑_(k=0)^n)(({{n}, {k}})) x y (((x-a k))^(k-1)) (((y-a (n-k)))^(-k+n-1))

Sum[Binomial[n,k]x y(((x-a k))^(k-1))(((y-a(n-k)))^(n-k-1)),{k,0,n}]

Subtract@@@Table[{(x+y)(((x+y-a n))^(n-1)),Sum[Binomial[n,k]x y(((x-a k))^(k-1))(((y-a(n-k)))^(n-k-1)),{k,0,n}]},{n,10}]//Simplify

(x^(-1)) (((x+y-a n))^n)==(∑_(k=0)^n)(({{n}, {k}}))  (((x-a k))^(k-1)) (((y-a (n-k)))^(-k+n-1))

Sum[Binomial[n,k](((x-a k))^(k-1))(((y-a(n-k)))^(n-k-1)),{k,0,n}]

Subtract@@@Table[{(x^(-1))(((x+y-a n))^n),Sum[Binomial[n,k](((x-a k))^(k-1))(((y-a(n-k)))^(n-k)),{k,0,n}]},{n,10}]//Simplify

(x^(-1)) (((x+y))^n)==(∑_(k=0)^n)(({{n}, {k}}))  (((x-a k))^(k-1)) (((y+a k))^(n-k))

Sum[Binomial[n,k](((x-a k))^(k-1))(((y+a k))^(n-k)),{k,0,n}]

Subtract@@@Table[{(x^(-1))(((x+y))^n),Sum[Binomial[n,k](((x-a k))^(k-1))(((y+a k))^(n-k)),{k,0,n}]},{n,10}]//Simplify