# Finding the # of paths in a grid from opposite corner but most avoid certain paths

In x-y coordinate you start out at (0,0) and you want to get to (8,14) by either moving up or right only.

You can't move to any points that are both odd, e.g. (1,1), (1,3)... (3,1), (3,3) ... (7, 13).

So my strategy is to count all the possible paths and subtract out the path that hits the (odd,odd) coordinates.

Total path in this 8 by 14 grid is

$$\binom{22}{8} or \binom{22}{14}$$

Now I need to subtract the paths that hit (odd,odd) from that.

I'm not wrapping my head around how to count the number of ways you can hit the (odd,odd) coordinates. One way is to brute force it by counting # of ways to get to (1,1), (1,3)... etc. but that's really inefficient and I think I will accidentally double count a lot of the paths.

Hint: In two consecutive steps, you must move uu (up, up) or rr (right, right). Let's shorten uu to U and rr to R. Then you have to make a total of $4$ R's and $7$ U's.
Therefore, the number of paths would be still the same if you delete all the rows and columns with red squares. The new grid has the size $4 \times 7$, resulting in $\binom{11}{4}$ possible paths.