gen-explain-counting-lattice-paths-as-you-would-to-a-capable-stu-7787

Explain counting lattice paths as you would to a capable student, in 3-4 sentences.

A lattice path is a route on a grid that moves from one point to another using only specific allowed steps, typically moving right or east and up or north. To count these paths, we use combinatorics: if you need to take *m* steps right and *n* steps up to reach your destination, you're essentially choosing which *m* positions out of your total (*m* + *n*) steps will be "right" moves, giving you $\binom{m+n}{m}$ total paths. This works because every valid path is uniquely determined by deciding when to step right versus up, and the order matters. For example, going from (0,0) to (2,3) requires 2 right steps and 3 up steps, so there are $\binom{5}{2} = 10$ different paths.