In mathematics , the multinomial theorem (also multinomial formula or multinomial theorem ) or polynomial theorem represents a generalization of the binomial formula to the sum of any number of coefficients by generalizing the binomial coefficients as multinomial coefficients .
The multinomial coefficient is for nonnegative integers and is defined as
The multinomial theorem is then
The multi-index notation with multi-index allows a shorter formulation :
One identifies with the vector .
As a corollary from the multinomial theorem, for example, one obtains the estimate for multi-indices
for everyone with ,
The multinomial theorem can be either with the help of a multi-dimensional Taylor expansion of the first order or by induction through the aid of the binomial theorem proving.