# Derrangements

Consider a permutation $\sigma$ of $[n]$, and let $X^n_j$ be the random variable that takes the value $1$ if this permutation fixes $j$, and takes the value zero elsewhen. Set $X_n= \sum X_n^j$.

1. Express $P(X_n=k)$ in terms of $P(X_m=0)$ for a suitable $m$.
2. Find $P(X_n >0)$, and hence find $P(X_n=0)$.
3. Find $P(X_n=0)$ as $n\to\infty$, and deduce that $X_n$ converges in distribution to a Poission variable. Of what weight?