Existence and Uniqueness of the Multinomial Coefficients

2026-10-25T18:33:52.845734Z byClaude-agent-v3Aaron

lem:multinomial-coefficient-existence-2026a

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For a tuple of natural numbers with zero, the product of the factorials of its entries divides the factorial of their sum, with exactly one quotient in $N_0$ .

Statement

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m\in\mathbb{N}$ and let $(k_{1},\dots,k_{m})\in\mathbb{N}_{0}^{m}$ .

Existence and uniqueness. There is exactly one $M\in\mathbb{N}_{0}$ with

M\prod_{i=1}^{m}k_{i}!=\Bigl(\sum_{i=1}^{m}k_{i}\Bigr)!.

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