For $k \in \llbracket 0, n \rrbracket$, show that the number of permutations of $\llbracket 1, n \rrbracket$ having exactly $k$ fixed points is $\binom{n}{k} d_{n-k}$. Deduce that $P_n\left(X_n = k\right) = \frac{d_{n-k}}{k!(n-k)!}$.
For $k \in \llbracket 0, n \rrbracket$, show that the number of permutations of $\llbracket 1, n \rrbracket$ having exactly $k$ fixed points is $\binom{n}{k} d_{n-k}$.
Deduce that $P_n\left(X_n = k\right) = \frac{d_{n-k}}{k!(n-k)!}$.