This is embarrasingly, the first problem in notes on introductory combinatorics by Polya and Tarjan. (Solved, but I havent looked).
Problem statement: Find the number of ways of spelling "abracadabra" always going from one letter to the adjacent.
$$A$$ $$B \quad B$$ $$R \quad R \quad R $$ $$A\quad A \quad A \quad A $$ $$C\quad C\quad C\quad C\quad C$$ $$A\quad A \quad A \quad A \quad A\quad A$$ $$D\quad D\quad D\quad D\quad D$$ $$A\quad A \quad A \quad A $$ $$B \quad B \quad B $$ $$R \quad R$$ $$A$$
I got a very improbable answer of $2^{25}$ so I tried a simpler case to understand it.
Starting at the northmost A there are two routes. At each of the two B's on the second row there are $2$ routes, so uptil this point
$$A$$ $$B \quad B$$ $$R \quad R \quad R $$
there should be $2^3$ ways to get the three letter word "ABR" but on manually counting the number of ways is just 4 (LL, LR, RR, RL where R=right/L=left). What is wrong with my approach? More precisely, where have I overcounted?
Edit: I understood my problem. I used the product rule instead of the sum rule. I think I will stop paying attention to these "rules" as they are hindering my problem solving anyway. (I have asked a previous questions on the subject)