TITLE: Is there a mathematical notation for restricting the depth of a factorial?
QUESTION [6 upvotes]: Say I only want to go $3$ "levels" deep for $5!$ and end up with just $5\times 4\times 3$. Is there a notation for this?

REPLY [14 votes]: This is also sometimes called "permutations of 3 out of 5 things" and denoted 5P3. You can find this on TI-8x calculators, at least, as nPr in the MATH->PRB menu (so you'd type 5 nPr 3).
I have also seen the falling factorial denoted  with brackets as $[5]_3$. I'd say any notation for this will have to be clarified when you use it, though.