HMMT 2003 E10

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?