Four Dice Sum to 20 Prob

This combinatorics question is about counting how many discrete outcomes produce a specific total when multiple random variables are each limited to a small finite set. It sits at the intersection of integer partitions and probability, a theme that appears constantly in quant prep material and in real interviews. You must reason about how a rigid constraint on the sum interacts with tight individual bounds, a structure that is simple to state but subtly restrictive.

It trains comfort with constrained counting, conditional probability and discrete distributions, plus the discipline to translate a worded setting into a clean combinatorial model. It also strengthens your ability to separate total sample space from favorable configurations, a key habit in rigorous quant interviews.

This matters because banks and hedge funds love quick, exact probability questions built on simple randomness models. Strong performance here signals the ability to handle payoff distributions, risk aggregation and scenario counting in more complex quant roles.

What it tests

Problems that ask for the number of ways to achieve a fixed sum with bounded integer variables (like dice faces) are governed by the principle of integer partitions with constraints. The core idea is that, while the sum is fixed, each variable (die) is limited to a finite range, so not all partitions are valid. The structure of the problem is to enumerate all possible combinations of values that meet both the sum and the bounds, then account for distinguishability (ordering) if required. This is a classic application of stars-and-bars with upper and lower bounds, but the constraints force you to check which partitions are feasible. The reason this works is that the total number of outcomes is a simple product, but the favorable outcomes require careful casework to respect the variable limits.