[Math] Which binds first, product or factorial

factorialnotationproducts

Which is the case:

$$
\prod_{i \in I}i! = \prod_{i \in I}(i!)
$$

or

$$
\prod_{i \in I}i! = \Bigg(\prod_{i \in I}i\Bigg)!
$$

Best Answer

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:

    1. Parenthesization,

    2. Factorial,

    3. Exponentiation,

    4. Multiplication and division,

    5. 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.

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