10. Bessie the cow is trying to navigate her way through a field. She can travel only from lattice point to adjacent lattice point, can turn only at lattice points, and can travel only to the east or north. (A lattice point is a point whose coordinates are both integers.) $(0,0)$ is the southwest corner of the field. $(5,5)$ is the northeast corner of the field. Due to large rocks, Bessie is unable to walk on the points $(1,1),(2,3)$, or $(3,2)$. How many ways are there for Bessie to travel from $(0,0)$ to $(5,5)$ under these constraints?
Solution:
32
In the figure, each point is labeled with the number of ways to reach that point. The numbers are successively computed as follows: The point $(0,0)$ can trivially be reached in 1 way. When Bessie reaches any subsequent point $(x, y)$ (other than a rock), she can arrive either via a northward or an eastward step, so the number of ways she can reach that point equals the number of ways of reaching $(x-1, y)$ plus the number of ways of reaching $(x, y-1)$. By iterating this calculation, we eventually find that $(5,5)$ can be reached in 32 ways.