# Interpret This Function First!

Calculus Level 5

$\large f( n ) =n!-\left( 1+\sum _{ r=1 }^{ n-2 }{ \left( \begin{matrix} n \\ r \end{matrix} \right) } f( n-r ) \right)$

Let $$f(n)$$ be a function defined for positive integers $$n \ge 2$$ such that the equation above is satisfied and $$f\left( 2 \right) =1$$.

Let $$\displaystyle m = \lim_ {n \to\infty} \dfrac{f(n)}{n!}$$, then find $$\left\lfloor 1000m \right\rfloor$$.

 Notation: $$\dbinom MN$$ denotes the binomial coefficient, $$\dbinom MN = \dfrac{M!}{N!(M-N)!}$$.

×