Item 2.
Consider all functions \(f: \{1,2,3,4,5\} \to \{1,2,3,4,5\}\text{.}\) How many of the injections have the property that \(f(x) \ne x\) for any \(x \in \{1,2,3,4,5\}\text{?}\)
Your friend claims that the answer is:
\begin{equation*}
5! - \left[ {5\choose 1}4! - {5 \choose 2}3! + {5\choose 3}2! - {5 \choose 4}1! + {5\choose 5}0! \right]\text{.}
\end{equation*}
Explain why this is correct.