Counting Positive Integer Triplets Summing to 25

This quant interview question is about counting how many ordered choices of positive integers can reach a fixed total. It sits at the intersection of combinatorics and discrete probability, a core piece of quant prep that appears across brainteasers, internship interviews, and full-time quant interviews. At top trading firms, this style of question is a quick way to check whether a candidate is comfortable with structured counting rather than ad hoc casework.

It trains your ability to model a verbal constraint as a clean combinatorial object, and to recognise when a problem hides a classic pattern from discrete math. You practice translating "positivity" conditions into a more convenient formulation and then using a standard counting toolkit for systematic enumeration.

This matters for quant interviews because many pricing, risk, and algorithm problems reduce to counting structured configurations under constraints. Interviewers want to see that you can represent a simple-seeming setup symbolically, generalise it, and reason about the size of the resulting state space without getting lost in manual enumeration. Mastering this kind of counting logic makes later probability, dynamic programming, and state-space modelling questions much more straightforward.

What it tests

When distributing a fixed total among several positive integer variables, the underlying structure is a partition of an integer into a set number of positive parts. The key insight is that requiring positivity (each variable at least 1) is equivalent to first allocating the minimum to each variable, then distributing the remainder freely. This transforms the problem into one of counting non-negative integer solutions, which is a classic application of the stars and bars theorem. The reason this works is that subtracting the minimum from each variable reduces the original constraint to a simpler, unrestricted form, making the combinatorial count tractable. The stars and bars method then counts the ways to distribute indistinguishable objects (the remainder) into distinguishable bins (the variables), capturing all possible ordered arrangements.