Functional Modulus

Algebra Level 5

\[|f(x_{1})-x_{1}|=|f(x_{2})-x_{2}|=|f(x_{3})-x_{3}|=\ldots=|f(x_{n})-x_{n}|\]

Let \(n\) be an odd positive integer and let \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, \ldots, x_{n}\) be distinct real numbers. Find the total number of one-to-one functions \(f: (x_{1}, x_{2}, x_{3},\ldots, x_{n}) \rightarrow (x_{1}, x_{2}, x_{3}, \ldots , x_{n})\) such that the equation above is satisfied.

This is not my original problem.
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