Suppose an a x b grid, and you start at the lower left corner and go to the upper right corner by moving only up and right. Suppose that a = 30, b = 40 and you cannot travel through the intersections (2, 3) and (12, 17). How many possible paths are there?
What my Approach was, and where I was stuck was:
number total number of possibilities without the exclusion, is
$a+b-2 \choose a-1$ = $30+40-2 \choose 30-1$ = $68 \choose 29$
but what I don't know is that how would I find the possibility where the point passes through that point, then I can just subtract those possibilities.
Good starting point and idea, inclusion and exclusion seems like the right tool. Let me build on what you think.
Guide:
To compute the number of path that passes through $(2,3)$, you just have to count the number of path from $(0,0)$ to $(2,3)$ and then multiply by the number of path from $(2,3)$ to $(30,40)$.
To compute the number of path that passes through $(12,17)$, you just have to count the number of path from $(0,0)$ to $(12,17)$ and then multiply by the number of path from $(12,17)$ to $(30,40)$.
To compute the number of path that passes through both $(2,3)$ and $(12,17)$, you just have to count the number of path from $(0,0)$ to $(2,3)$ and then multiply by the number of path from $(2,3)$ to $(12,17)$ and then multiply by the number of path from $(12,17)$ to $(30,40)$.