Label each point with the number of paths to get to that point.

Are we only allowed to travel east and north (right and up)? Presumably this is the case since otherwise the answer is infinitely many paths.

Now along the bottom edge there are all 1's since you can only get to each point by a move right from the previous point, same with the left edge.

Continue to fill in the grid. (HINT: the number of paths to a pointpis the sum of the number of paths to the two points which can lead top)

If you must pass through pointCjust break the problem into 2 subproblems, the number of paths fromAtoCand the number of paths fromCtoB. With that information what is the total number of paths fromAtoBthat pass throughC?

An interesting class of such problems requires the grid to be a square and asks for the number of paths from A to B that don't cross the diagonal (stay in the lower triangular half of the square).