Suppose I have to go from $A$ to $I$ in such a way that I should not visit one tile more than once, in my path, and only horizontal and vertical movements are allowed. I brute forced the solution and found that there are $12$ possible ways to reach $I$ starting from $A$. I am wondering if there is any combinatorics method to solve the problem.
My analysis is that every tile except $E$ has $1$ input and $2$ possible outputs. $E$ has $1$ input $3$ possible outputs. So, all possible paths from $A$ will be $(2^7) + (3^1) = 131$. Thus, I am over-counting.