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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