Going on a grid

For a simple grid, we can solve the problem by counting the actual number ways as follows.

1. Assign the column numbers $x = 0, 1, 2, 3, 4$ and row numbers $y = 0, 1, 2, 3$ starting from the destination (movie).
2. Enter the numbers of ways $n(x,y)$ (shown in blue) to the destination starting from $x=0$ and $y=0$ . When $x=0$ and $y=0$ , there is only 1 way to the destination, therefore, $n(0, y) = n(x, 0) = 1$ . It is obvious that $n(1,1) = n(1,0) + n(0,1)$ and in general $n(x,y) = n(x, y-1) + n(x-1,y)$ .

Following the process we find that $n(4,3) = \boxed{35}$ . By observation, it is found that $\displaystyle n(x,y) = {x+y \choose y} = {x+y \choose x}$