Find the number of routes from leftmost vertex to rightmost vertex 

Suppose that one can move from one vertex to another if, and only if,
  the two vertices are connected by a unique common edge. The number of
  routes that one can take from the leftmost vertex L through 6 edges
  and 5 intermediate vertices to the rightmost vertex R is .......

This is my scholarship practice exam (pre-university level) and the only way I did it correctly is by counting but I feel it was a bit messy but it can be simplified by writing with alphabets such as (u for up, d for down, s for go straight). 
However, there could be other solutions such as combination or permutation, probably. The answer key provided is 12. Please advise what you think and how to solve this problem.

Try labeling the vertices and writing down the resulting adjacency matrix. Then raise it to the power of 6, and read the entry in the (i, j) position, where i and j are the labels of the left- and right- most vertices, respectively. Since the adjacency matrix is 14 x 14, you should probably use a computer. I recommend using the Wolfram alpha website. Although it wouldn't be too hard by hand, since A^6 = (A^2)^2 A^2.
