Where Should I Open My Restaurant?

We answer a question in real estate that is a variation of a problem that arises in many combinatorics and discrete mathematics courses. There are three things that matter in property: location, location, location. Unknown. Possibly Lord Samuel of Britain [8] There is a lot of truth in this opening quote, especially applied to commercial real estate. If you want your restaurant to succeed, customers have to walk through the door. 1 The Problem A walk on the Z lattice using only the steps (0, 1), which we call ‘North’, and (1, 0), which we call ‘East’, is a lattice path. Question 1. 1. How many lattice paths begin at (0, 0) and end at (m,n)? 2. What proportion of these paths pass through the point (a, b)? Versions of these elementary exercises appear in many introductory combinatorics textbooks, as lattice paths provide a nice setting to explore identities involving binomial coefficients. See for example [3, Chapter 4], [2, Section 3.5], [4, Section 8.5], and [5, Section 2.6]. Solution. 1. A path from (0, 0) to (m,n) consists of m steps East, and n steps North, and can be uniquely identified with a word of length m + n consisting of m E’s and n N’s. Therefore, there are ( m+n m ) total lattice paths.