How many lattice paths with step $S$ and $W$ are there that begin at $(0,0)$, end at $(-12,-12)$ and do not go through any of the points $(-1,-4), \space (-5,-3), \space (-9,-11)$?
I'm unsure of how to approach this question. Any hint would be good. Thank you.
Let $c_1$ = the set of paths going through (-1,-4), $c_2$ = the set of paths going through (-5,-3), etc.
Then, use the inclusion-exclusion principle to solve for N($c_1'c_2'c_3'$).
$N$ = $24\choose 12$.
$N(c_1)$ = $5\choose 4$$19\choose 11$ etc.