Multinomial Theorem
Definition: The Multinomial Theorem is an extension of the Binomial Theorem and is a fundamental concept in combinatorics and algebra. It provides a formula for expanding expressions involving multiple terms, each raised to different positive integer exponents. The theorem is particularly useful when dealing with polynomials with more than two terms.
Key Concepts:

Generalized Binomial Theorem: The Multinomial Theorem generalizes the Binomial Theorem, which is used to expand expressions of the form $(a+b{)}^{n}$. The Multinomial Theorem deals with expressions of the form $({a}_{1}+{a}_{2}+{a}_{3}+\dots +{a}_{k}{)}^{n}$, where there are "k" terms, each raised to the power of "n."

Coefficients and Exponents: In the expansion, the coefficients of each term are determined by a multinomial coefficient, while the exponents of the individual terms are based on the exponents of the original terms. The multinomial coefficient is calculated using combinations and accounts for the ways terms can be grouped together.

Multinomial Coefficient: The multinomial coefficient $\left(\genfrac{}{}{0px}{}{n}{{n}_{1},{n}_{2},{n}_{3},\dots ,{n}_{k}}\right)$ represents the number of ways to arrange "n" objects into "k" distinct groups, where ${n}_{1}$ objects are in the first group, ${n}_{2}$in the second, and so on, with ${n}_{k}$in the last group. It is calculated as: $\left(\genfrac{}{}{0px}{}{n}{{n}_{1},{n}_{2},{n}_{3},\dots ,{n}_{k}}\right)=\frac{n!}{{n}_{1}!\cdot {n}_{2}!\cdot {n}_{3}!\cdot \dots \cdot {n}_{k}!}$

Formula for the Multinomial Expansion: The Multinomial Theorem provides a formula for expanding expressions of the form $({a}_{1}+{a}_{2}+{a}_{3}+\dots +{a}_{k}{)}^{n}$: $({a}_{1}+{a}_{2}+{a}_{3}+\dots +{a}_{k}{)}^{n}=\sum \left(\genfrac{}{}{0px}{}{n}{{n}_{1},{n}_{2},{n}_{3},\dots ,{n}_{k}}\right)\cdot ({a}_{1}^{{n}_{1}}\cdot {a}_{2}^{{n}_{2}}\cdot {a}_{3}^{{n}_{3}}\cdot \dots \cdot {a}_{k}^{{n}_{k}})$

Example: Suppose you want to expand $(x+y+z{)}^{3}$. The expansion is: $(x+y+z{)}^{3}=\left(\genfrac{}{}{0px}{}{3}{3,0,0}\right){x}^{3}+\left(\genfrac{}{}{0px}{}{3}{2,1,0}\right){x}^{2}y+\left(\genfrac{}{}{0px}{}{3}{1,2,0}\right)x{y}^{2}+\left(\genfrac{}{}{0px}{}{3}{0,3,0}\right){y}^{3}+\left(\genfrac{}{}{0px}{}{3}{2,0,1}\right){x}^{2}z+\left(\genfrac{}{}{0px}{}{3}{1,1,1}\right)xyz+\left(\genfrac{}{}{0px}{}{3}{0,2,1}\right){y}^{2}z+\left(\genfrac{}{}{0px}{}{3}{1,0,2}\right)x{z}^{2}+\left(\genfrac{}{}{0px}{}{3}{0,1,2}\right)y{z}^{2}+\left(\genfrac{}{}{0px}{}{3}{0,0,3}\right){z}^{3}$

Applications:
 The Multinomial Theorem is applied in various fields of mathematics, physics, and engineering when dealing with polynomials involving multiple variables and exponents.
 It is used in probability and statistics to calculate probabilities in multivariate distributions.