Definition:Binomial Coefficient/Multiindices

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Definition

Let $k = \sequence {k_j}_{j \mathop \in J}$ and $\ell = \sequence {\ell_j}_{j \mathop \in J}$ be multiindices.

Let $\ell \le k$.

Then $\dbinom k \ell$ is defined as:

$\ds \binom k \ell = \prod_{j \mathop \in J} \binom {k_j} {\ell_j}$

Note that since by definition only finitely many of the $k_j$ are non-zero, the product in the definition of $\dbinom k \ell$ is convergent.