\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Let \(\alpha, \beta \in \mathbb{N}^n\) multi-indices. Then

\begin{equation*} \binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1} \cdots \binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!}. \end{equation*}

The difference is meant entry-wise.

See also Link to heading

• upper estimate