# Subfactorial

Number Theory Level pending

Find a six-digit number represented as $$\overline{abcedf}$$ that has the following property:

$\overline{abcdef} = !a + !b + !c + !d + !e + !f$

The exclamation mark in front of the number indicates a sub-factorial which is defined as follows:

$!n = n! \left(1 - \frac 1{1!} + \frac 1{2!} - \frac 1{3!} + \cdots + (-1)^n \frac 1{n!} \right)$

For examples: $$!2=1$$ and $$!6=265$$

The exclamation mark after the number denotes the factorial function; for example $$8! = 1 \times 2 \times 3 \times \cdots \times 8$$.

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