Fixed Points of a Permutation

Use indicator variables $I_i$ for object $i$ being fixed.

By linearity of expectation, sum the individual probabilities $P(I_i=1)=1/n$.

1. 1. Define indicators

   Let $I_i=1$ if object $i$ is in its original position, else 0. Then the number of fixed points is $X={\sum }_{i=1}^n{I}_i$.

2. 2. Individual probability

   By symmetry, $P(I_i=1)=1/n$ for each $i$.

3. 3. Linearity of expectation

   $E[X]={\sum }_{i=1}^{n}E[I_i]=n\cdot \frac{1}{n}=1$.

   $$E[X]=1$$

Answer: 1