HMMT November 2022 U13

Consider the paths from $(0,0)$ to $(6,3)$ that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the $x$-axis, and the line $x=6$ over all such paths. (In particular, the path from $(0,0)$ to $(6,0)$ to $(6,3)$ corresponds to an area of 0 .)