Navigational Help combinations You take exactly 10 steps.  You can move in one of three directions (North West East) with no desired target point (just 10 steps).  You cannot follow a move East with a move West or vice versa.  How many possible combinations of paths are there.  I know it's 3^10 but am unsure as to how to limit the counter from the repetition of only two particular instances 
 A: We want to count the number of "words" of length $10$ made up of the letters N and/or W and/or E, with the rule that W cannot be immediately followed by E, and E cannot be immediately followed by W. Briefly, call a word in which W is not immediately followed by E, or the other way around, a good word. 
Let $w_n$ be the number of good words of length $n$. It is easy to see that $w_1=3$. Careful listing shows that $w_2=7$. (One might want to start counting with the good words of length $0$. There is $1$ such word, the empty word, so $w_0=1$.)
Now let us look at $n\ge 3$. There are two types of good word of length $n$: (i) the ones that end in NN and (ii) the ones that don't.
(i): There are $w_{n-2}$ good words of length $n$ that end in NN, for to get such a word we just append NN to any good word of length $n-2$. 
(ii) There are $2w_{n-1}$ good words of length $n$ that don't end in NN. For  take any good word of length $n-1$. If it ends in N, it can be completed in $2$ ways into a word that doesn't end in NN (append W or E). If it ends in W, again it can be completed in $2$ ways (append N or W). Also, it it ends in E it can be completed in $2$ ways. We have reached the recurrence formula
$$w_{n}=2w_{n-1}+w_{n-2}.\tag{1}$$
Now we can start computing. Since $w_2=7$ and $w_1=3$, by (1) we get $w_3=2\cdot 7+2=17$. But then, applying (1) again, we get $w_4=41$. But then applying (1) again, we get $w_5=99$. It is not hard to continue and find $w_{10}$.
Remark: The recurrence (1) can be used to find a closed form explicit formula for $w_n$. I believe you are not being asked to do that, so will omit the derivation of a general formula. 
