The question is just what you would expect from the title:
How many paths (with steps going North or East) are there from $(0,0)$ to $(4,6)$, where three consecutive steps in any direction are prohibited?
My attempted solution with a generating function
We can consider distributing northward steps between the four eastward steps: there are five "slots" for the north steps to be situated, and we can have either $0,1,2$ consecutive north steps. This gives the answer $[x^6](1+x+x^2)^5 = 45$.
I would like to learn how to solve this using inclusion exclusion for good practice and possibly to verify the above result. My issue in particular is how to break apart the cases.
Edit to correct the generating function solution
As pointed out in the comments, the generating function solution still allows three right steps. However, there are only two ways this happens, namely
$$ \begin{align} &\underline{uu}r\underline{uu}rrr\underline{uu},\\ &\underline{uu}rrr\underline{uu}r\underline{uu}, \end{align} $$
so the right answer is $45-2=43$.
