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Good functional and set problem

Source: 2009 Jozsef Wildt International Math Competition

April 17, 2020
Setsfunction

Problem Statement

Let consider the following function set F={f ∣ f:{1, 2, ⋯ , n}→{1, 2, ⋯ , n}}F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}
[*] Find ∣F∣|F| [*] For n=2kn=2k prove that ∣F∣<e(4k)k|F|< e{(4k)}^{k} [*] Find nn, if ∣F∣=540|F|=540 and n=2kn=2k
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