Let $F_n$ be the random variable that is the number of fixed points of a random permutation on $n$ elements. As $n \to \infty$, the distribution of $F_n$ approaches a Poisson distribution with mean 1. This can easily be shown via direct calculation.
I would like a more intutive proof. For large $n$ the events that any particular $k << n$ elements are fixed are approximately independent. If we are assume they are exactly independent, it is immediate that the limit is Poisson with mean 1. Is there a way to make this reasoning precise, ideally with a minimum of calculation?