Five Fixed Points in Permutations

This quant interview question is about the asymptotic distribution of the number of fixed points in a random permutation, a classic example in discrete random variables and combinatorial probability. It sits at the intersection of combinatorics and limit theorems, which are central themes in serious quant prep for front-office interviews. You are asked to extract a clean limiting probability from a messy finite-n combinatorial structure.

It trains your understanding of rare events, indicator random variables, and convergence in distribution, especially to the Poisson law. It also reinforces your ability to move between exact finite models and elegant asymptotic descriptions, and to recognize when complex dependence can be safely ignored. This is core material in advanced probability for quant interviews.

This matters because modern quantitative finance heavily uses Poisson approximations, point processes, and limit theorems in modeling order arrivals, jump processes, and risk events. Interviewers use questions like this to see if you can handle abstraction, asymptotics, and probabilistic reasoning beyond rote formulas. For highly competitive quant interviews, being fluent with this type of argument is a strong differentiator in your quant prep.

What it tests

When analyzing the number of occurrences of a rare event in a large set—such as fixed points in a random permutation—the underlying structure is that each position has a small, nearly independent chance of exhibiting the event (here, being a fixed point). As the number of trials grows and the event probability shrinks so that their product remains constant, the sum of these indicators converges in distribution to a Poisson random variable. This is formalized by the Poisson Limit Theorem, which explains why, despite slight dependencies, the count of such rare events behaves like a Poisson process. The key is that the dependencies between events become negligible as the set size increases, making the Poisson model an excellent approximation. This principle is powerful because it allows us to replace complex combinatorial dependencies with a simple, universal distribution in the limit.