Three Dice Sum to 18

This quant interview question lives at the intersection of combinatorics and probability, wrapped around order statistics for discrete random variables. You are dealing with several random outcomes and focusing only on the largest ones, under a tight constraint on their total. That makes it a neat test of whether you can translate an intuitive description into a precise probabilistic event, which is central in quant prep for trading and research roles.

It trains conditional probability, counting methods, and sample-space reasoning under structural constraints. You have to recognize when an apparently wide set of possibilities collapses to a very small family of feasible configurations. The exercise is about spotting that structure quickly, encoding it correctly, and then doing clean combinatorial counting without missing or double-counting scenarios.

This matters in quant interviews because many real models depend on extremes: worst losses, best returns, highest exposures, tail events. Interviewers want to see that you can move fluidly from a verbal description of an extreme-condition event to a compact, exact probability. It showcases your ability to simplify complex setups, reason rigorously about dependence induced by ordering, and compute efficiently under pressure, all of which are core skills for quantitative finance roles and effective quant prep.

What it tests

When a problem asks about the sum of the largest (or smallest) $k$ values out of $n$ independent, identically distributed random variables, the key is to translate the sum constraint into specific value constraints on the variables. Often, the only way to achieve an extreme sum is for those $k$ variables to all take the maximum (or minimum) possible value, which sharply restricts the configurations you need to count. This principle holds because the sum of the largest $k$ values is maximized only when each of them is maximized, and any reduction in any of these values would reduce the sum below the target. Recognizing this allows you to enumerate cases based on how many variables achieve the maximum, and then count the number of ways the remaining variables can take on lesser values. The structure is combinatorial: identify which variables must be maximal, then count the arrangements of the rest.