Question: 2015 Fall Final - 18

The $n$ students $S_1,S_2,\dots,S_n$ decide to organize a Secret Santa: They take a uniformly random permutation $P_1,P_2,\dots,P_n$ of $S_1,S_2,\dots,S_n$. For each $i = 1,2,\dots,n$, student $S_i$ buys a gift and gives it, anonymously, to student $P_i$.

Let $X$ be the number of students who give their gift to themselves. What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
Hint: Use an indicator random variable for each student.