At Least One Correct Room

This question considers a random reassignment of many distinct people to many distinct rooms, with each person originally having a designated room. After their room keys are all collected and randomly redistributed, everyone simply goes to the room matching the key they hold. The focus is on the chance that, under this completely random matching, at least one person ends up in the exact room they were originally assigned. It is a canonical probability puzzle about large random systems and is frequently seen in interviews for quantitative trading and research roles, especially where familiarity with classical discrete probability is expected.

The underlying ideas center on random permutations, fixed points, and how these behave as the system size grows. A candidate is expected to recognize and exploit the symmetry of the situation, use inclusion–exclusion in a controlled way, and connect the combinatorial expressions to limiting behavior involving the exponential function. Interviewers watch for comfort with factorial-scale counting, the ability to manage overlapping events systematically, and clarity in moving from finite formulas to clean asymptotic reasoning without handwaving.

What it tests

This problem class is governed by the structure of random permutations and the probability of fixed points, which are instances where an object returns to its original position after shuffling. The key insight is that, for large $N$, the probability that at least one object remains in its original place after a random shuffle converges to a constant, independent of $N$. This is because the number of ways to permute $N$ objects with no fixed points (derangements) grows factorially, but the ratio of derangements to total permutations approaches $e^{-1}$ as $N$ increases. The inclusion-exclusion principle quantifies the overlap between events (here, guests getting their own keys), and its alternating sum structure naturally leads to the exponential function in the limit. The reason this pattern holds is that the chance of any specific object being fixed is $1/N$, but the dependencies between multiple fixed points are captured exactly by inclusion-exclusion, and their sum stabilizes as $N$ grows.