Powerful Limit

Calculus Level 5

$\large f(n) = \sum_{r=0}^{n} \dbinom{n}{r} r^r (2-r)^{n-r}$

Find the value of the closed form of $$\displaystyle \text{L} = \lim_{n\to\infty} \dfrac{f(n)}{n!}$$.

Submit your answer as $$\lfloor 1000 \text{L} \rfloor$$.

You may use a calculator for the final step of your calculation.

Details and assumptions:

• Assume that we adopt the convention that $$0^0 = 1$$.
• $$\lfloor \cdot \rfloor$$ denotes the floor function.
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