gen-define-counting-lattice-paths-and-give-one-short-worked-exam-f50b

Define counting lattice paths and give one short worked example.

# Counting Lattice Paths ## Definition A **lattice path** is a sequence of steps on a grid (typically the integer lattice) moving from a starting point to an ending point, where each step moves along grid lines in specified directions (commonly right R or up U). **Counting lattice paths** means determining how many distinct paths exist between two points following given movement rules. For paths from (0,0) to (m,n) using only right (R) and up (U) steps, the number of paths is: $$\binom{m+n}{m} = \binom{m+n}{n} = \frac{(m+n)!}{m! \cdot n!}$$ ## Worked Example **Problem:** How many lattice paths go from (0,0) to (2,3) using only right and up steps? **Solution:** - We need exactly 2 right steps (R) and 3 up steps (U) - Total steps needed: 2 + 3 = 5 - We must choose which 2 of the 5 positions are R steps (or equivalently, which 3 are U steps) $$\binom{5}{2} = \frac{5!}{2! \cdot 3!} = \frac{5 \times 4}{2 \times 1} = 10$$ **Answer:** There are **10 distinct paths**. One such path example: RRUUU (right, right, up, up, up)