Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example:
$$ 3 = 2! + 1!$$ $$ 4 = 3! - 2! = 2! + 1! + 0!$$ $$ 146 = 4!\cdot3! + 2!$$
etc. My gut instinct says that this isn't true, but I can't see why. Something like 8076 doesn't have an obvious solution, but maybe you can get it by subtraction a huge factorial from the product of two smaller factorials or something. Or maybe there's a way of finding sets of factorials that add/subtract/multiply to 1, in which case any number can be constructed this way. I've tried finding something but haven't had much luck.
EDIT: Oops, positive integer, not positive number.