Solve without a calculator– 2

First we approximate the given quotient by $\frac{5!}{e} = \frac{120}{e} < \frac{120}{2.718} < \frac{120}{2.7} = \frac{400}{9} = 44.\bar{4}$

Using $[\cdot]$ to denote the nearest integer function and $!n$ to denote the subfactorial, the above inequality implies $44 = {}!5 = \left[\frac{5!}{e}\right] \leq \left[\frac{120}{2.718}\right] \leq \left[44.\bar{4}\right] = 44$ so the answer is $\boxed{44}\text{.}$

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Notes: Obviously one can just solve this with a piece of paper and pencil [y'know, like before calculators?]. Due to the numbers chosen, however, it's interesting to give a solution which is more combinatorial in nature. Also, you might have noticed I didn't include the nontrivial step of evaluating $!5\text{,}$ but anyone interested can evaluate it directly by using the recursive property $!n = n\cdot {}!(n-1) + (-1)^n \qquad !1 = 0$

$\dfrac {120}{2.718} = \dfrac {1200}{27.18}$ which nearly equals. $\dfrac {1200}{27} = \dfrac {400}{9} = 400\times \dfrac {1}{9} = 400\times 0.1111..... = 44.44444.......$

$ANSWER:[44.44...] = \boxed { 44 }$