Definition:Multinomial Coefficient

Definition

Let $k_1, k_2, \ldots, k_m \in \Z_{\ge 0}$ be positive integers.

The multinomial coefficient of $k_1, \ldots, k_m$ is defined as:

$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} := \dfrac {\left({k_1 + k_2 + \cdots + k_m}\right)!} {k_1! \, k_2! \, \ldots k_m!}$

Trinomial Coefficient

The trinomial coefficient of $k_1, k_2, k_3$ is a particular case of a multinomial coefficient, defined as:

$\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} := \dfrac {\left({k_1 + k_2 + k_3}\right)!} {k_1! \, k_2! \, k_3!}$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(41)$