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, \( \dbinom MN = \dfrac{M!}{N!(M-N)!} \).

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