Lesson 6

Problem. How many fixed points are there on average in a random permutation of numbers \([n]\)? (More formally, the problem is stated as follows. Consider a random permutation \(\pi\) of numbers \([n]\) and let \(\alpha=|\{i \in [n] \colon \pi(i)=i\}|\) — the random variable equal to the number of fixed points in permutation \(\pi\). Find \(\operatorname{E}[\alpha]\).)