To get from A to C we have this possible path

right up up

But we could select the "ups" to be in any two of the three positions

So C(3,2) = 3 possible paths from A to C

To get from C to B we have this possible path

right right right right up up

But we could select the "ups" to occupy any two of the six positions

So C(6,2) = 15 possible paths from C to B

So...the total possible paths from A to B = 3 * 15 = 45