It's not clear what's you logic behind applying permutation to this problem. I guess 3 is # of the forks on the road, but we are not choosing those 3 out of 7 (?).
Anyway, the problem is about simple counting. There are 3 forks in the road: 2 choices for the first one, 2 for the second and 3 for the third. Hence total # of ways is 2*2*3=12.
7 paths to chose from and you must pick 3 to get to the destination..
I haven't seen a permutation problem yet in the OG, so maybe the difference in identifying when to use it will show when I see one.