Definition:Binomial Coefficient/Multiindices

Definition
Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \mathop \in J}$ be multiindices.

Let $\ell \le k$.

Then $\displaystyle \binom k \ell$ is defined as:
 * $\displaystyle \binom k \ell = \prod_{j \mathop \in J} \binom {k_j} {\ell_j}$

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