Be a reverse engineer

Algebra Level 5

\[{n \choose 0} - \frac 12 {n \choose 1} + \frac 13 {n \choose 2} - \ldots + (-1)^n \frac 1{n+1} {n \choose n} = 0.00125 \]

What is the value of \(n\) that satisfy the equation above?

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