4. (7 points) We want to count the number of paths from vertices $m$ to $n$ in the following graph. Each path is made up of a series of steps, where each step is a move one unit to the right or a move one unit upward (No moves to the left or downward are allowed.)

The graph is a grid of unit squares spanning 6 columns and 4 rows (i.e., lattice points from $(0,0)$ to $(6,4)$). Vertex $m$ is located at the bottom-left corner $(0,0)$, and vertex $n$ is located at the top-right corner $(6,4)$. Vertex $x$ is located at $(3,2)$ (3 units to the right and 2 units up from $m$).

a. (3 points) What is the number of paths from vertices $m$ to $n$?

b. (4 points) What is the number of paths from vertices $m$ to $n$ that do not go through vertex $x$?