Multinomial theorem

The multinomial theorem expands a nonnegative integer power of a finite sum into terms indexed by exponent choices, with multinomial coefficients giving their multiplicities.

Multinomial coefficient: A coefficient counting arrangements of a multiset, given by a factorial quotient. It gives the coefficient of each term in the theorem’s expansion.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and nonnegative integer powers. The theorem is an identity between polynomials.
Multiset permutation: An ordering of objects drawn from a collection containing repeated, indistinguishable elements. Each coefficient counts distinct orderings of the chosen summands.
Isaac Newton: An English mathematician and natural philosopher who developed foundational work in calculus, mechanics, and optics. Newton’s generalized binomial work helped establish broader power-expansion methods.
Binomial theorem: A formula expanding an integer power of a sum of two terms. It is the two-term special case of the multinomial theorem.
Exponentiation: Repeated multiplication of a base by itself, defined for integer exponents through algebraic rules. The theorem rewrites repeated multiplication of a sum.
Multinomial test: A statistical test comparing observed category counts with specified probabilities. Its likelihood calculations use multinomial coefficients for category-count outcomes.
James Gregory: A seventeenth-century Scottish mathematician known for work on infinite series and geometry. Gregory independently developed generalized binomial expansions relevant to the theorem’s history.
Stars and bars: A method for counting nonnegative integer solutions to equations whose variables sum to a fixed total. It counts the exponent vectors indexing terms when the number of summands is fixed.
Factorial: For a nonnegative integer, the product of all positive integers up to that integer, with zero factorial equal to one. Factorials express the theorem’s multinomial coefficients.

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