Multinomial Coefficient

Author

Eric W. Weisstein
March 4, 2005

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

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

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

Definition

multinomial[n__]:=(((Plus@@{n})!)/(Times@@(#!&/@{n})))

FullSimplify[multinomial[k,n-k]==Multinomial[k,n-k]]

Special Cases

Binomial

FullSimplify[Multinomial[k,n-k]]

LeafCount/@{Multinomial[k,n-k],Binomial[n,k]}

FullSimplify[FunctionExpand[Multinomial[k,n-k]]]

LeafCount/@{Multinomial[k,n-k],((Gamma[1+n])/(Gamma[1+k] Gamma[1-k+n]))}

FullSimplify[Multinomial[k,n-k]-Binomial[n,k]]

Trinomial

Table[Multinomial[n1,n2,n3],{n1,0,7},{n2,0,n1},{n3,0,n2}]

FunctionExpand[Multinomial[n1,n2,n3]]

Identities

FullSimplify[Subtract@@{
Multinomial[n1,n2,n3,n4,n5],
Multinomial[n1+n2,n3,n4,n5]Multinomial[n1,n2]
}]

FullSimplify[Subtract@@{
Multinomial[n1,n2,n3,n4,n5],
Multinomial[n1+n2+n3,n4,n5]Multinomial[n1,n2,n3]
}]