TITLE: Which binds first, product or factorial?
QUESTION [10 upvotes]: Which is the case:
$$
\prod_{i \in I}i! = \prod_{i \in I}(i!)
$$
or
$$
\prod_{i \in I}i! = \Bigg(\prod_{i \in I}i\Bigg)!
$$

REPLY [14 votes]: The convention 
  \begin{align*}
\prod_{i \in I}i! = \prod_{i \in I}(i!)\tag{1}
\end{align*}
  is also affirmed by the operator precedence rules stated in OEIS.

For standard arithmetic, operator precedence is as follows:

Parenthesization,
Factorial,
Exponentiation,
Multiplication and division,
Addition and subtraction.



and since the product sign $\prod$ is just a short-hand for successively using the multiplication operator, the convention (1) is valid.