First recognize that, in order to get from point X to point Y, we MUST travel through points A,B,C,D,E and F.

So, we can take the task of getting from point X to Y and break it into stages.

Stage 1: Move from point X to point A
There's only 1 possible route, so we can complete stage 1 in 1 way.

Stage 2: Move from point A to point B
There are 2 possible routes, so we can complete stage 2 in 2 ways.

Stage 3: Move from point B to point C
There's only 1 possible route, so we can complete stage 3 in 1 way.

Stage 4: Move from point C to point D
There are 2 possible routes, so we can complete stage 4 in 2 ways.

Stage 5: Move from point D to point E
There's only 1 possible route, so we can complete stage 5 in 1 way.

Stage 6: Move from point E to point F
There are 3 possible routes, so we can complete stage 6 in 3 ways.

Stage 7: Move from point F to point Y
There's only 1 possible route, so we can complete stage 7 in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus move from point X to point Y) in (1)(2)(1)(2)(1)(3)(1) ways (= 12 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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