How many paths are there from A to B passing through C, if every step must be up or to the right?
See the answer here:
From A to C we have all the possible arrangements of this set
( North, North, East) = C(3,1) = 3
From C to B we have all the possible arrangements of this set
( North , North, East, East, East, East ) = C(6,2) = 15
So...the total number of paths from A to B through C are 3 * 15 = 45