Fix some `n`. Suppose we choose a permutation on `n` elements uniformly and at random. What is its average number of fixed points?

Group theory classes usually have some convoluted argument which establishes this result -- usually using the fixed point formula for a finite group acting on a finite set. Here's a very simple probability theory argument.

Take some `i`. What is the probability that a randomly-chosen permutation fixes `i`? A little thought shows that it's 1/`n` -- for instance, you can just consider the symmetry. Define random variables:

**N** = number of fixed points
**N**_{i} = 1 if `i` is a fixed point, otherwise.

Then

**N** is the

sum of all the

**N**_{i}:

**N** = ∑_{i=1,...,n} **N**_{i}.

The expected value of **N**_{i} is just the probability of `i` being a fixed point, or 1/`n`. Since expectation is linear, the average value of **N** is the sum of the average values of the **N**_{i}. This is a sum of `n` times 1/`n`, or just 1.

**QED**.