Grid Paths with Forced Detour

This combinatorics question is about counting constrained paths on a grid when you must visit a specific intermediate location before reaching the final target. It sits at the intersection of discrete math and probability ideas, which are core to quant prep and many technical interviews. The twist is the forced detour: paths are not just from start to finish, but must pass through a mandatory checkpoint.

It trains your ability to decompose a complex counting problem into independent stages linked by a constraint. You need to recognize structure, formalize it, and translate a geometric or game-like description into clean combinatorial quantities. This is exactly the kind of mental flexibility that strong performance in quant interviews requires.

For quant interviews, this matters because path counting under constraints underlies lattice models, dynamic programming, and scenario trees in trading and risk. It tests whether you can reason cleanly about state spaces, intermediate states, and how local choices compose into global outcomes, a key skill in quantitative research and algorithm design.

What it tests

Whenever a path or sequence must pass through a specific intermediate point, the total number of valid paths is the product of the number of ways to reach that point and the number of ways to continue from it to the destination. This is because, under independence, every way of reaching the intermediate point can be paired with every way of continuing from it. The key is that the constraints before and after the intermediate point are independent except for the requirement to pass through that point. This approach generalizes to any problem where a journey or process must visit certain checkpoints in a fixed order, and the total count is the product of the counts for each segment. The principle holds because the choices made in each segment are disjoint and collectively exhaust all possibilities that satisfy the constraint.