# Multinomial theorem

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 .

## formula

The multinomial coefficient is for nonnegative integers and is defined as ${\ displaystyle k_ {1}, \ ldots, k_ {n}}$${\ displaystyle k: = \! \, k_ {1} + \ ldots + k_ {n}}$

${\ displaystyle {k \ choose k_ {1}, \, \ ldots, \, k_ {n}}: = {\ frac {k!} {k_ {1}! \ cdot \, \ ldots \, \ cdot k_ {n}!}}.}$

The multinomial theorem is then

${\ displaystyle (x_ {1} + x_ {2} + \ ldots + x_ {n}) ^ {k} \, = \ sum _ {k_ {1} + \ ldots + k_ {n} = k} {k \ choose k_ {1}, \ ldots, k_ {n}} \, \ cdot \, x_ {1} ^ {k_ {1}} \ cdot x_ {2} ^ {k_ {2}} \ cdots x_ {n } ^ {k_ {n}}.}$

The multi-index notation with multi-index allows a shorter formulation : ${\ displaystyle \ alpha}$

${\ displaystyle (x_ {1} + x_ {2} + \ cdots + x_ {n}) ^ {k} = \ sum _ {| \ alpha | = k} {{k} \ choose \ alpha} \ cdot x ^ {\ alpha}}$

One identifies with the vector . ${\ displaystyle x}$${\ displaystyle (x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {R} ^ {n}}$

## application

As a corollary from the multinomial theorem, for example, one obtains the estimate for multi-indices

${\ displaystyle n ^ {k} = (1+ \ cdots +1) ^ {k} = \ sum _ {| \ beta | = k} {\ frac {| \ beta |!} {\ beta!}} \ geq {\ frac {| \ alpha |!} {\ alpha!}}}$for everyone with ,${\ displaystyle \ alpha}$${\ displaystyle | \ alpha | = k}$

so

${\ displaystyle | \ alpha |! \ leq n ^ {| \ alpha |} \ cdot \ alpha!}$.

## Evidence sketch

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. ${\ displaystyle n}$