Odd North-Eastern Paths

Look at this $7 \times 12$ grid. We have to go from $(0,0)$ to $(7,12)$ without going through odd-odd lattice points, which have been represented by dots. It is not possible to take exactly an odd number of steps in one direction and then change direction. In effect, we must take 2 steps to the right or 2 steps to the up at once. Denote 2 steps towards the east by $R$ and 2 steps towards the north by $U.$ Since we have to go from $(0,0)$ to $(7,12)$, we must have a string of 3 $R$s and 6 $U$s.

In fact, for every string of 3 $R$s and 6 $U$s, we have one path from $(0,0)$ to $(7,12).$ And for every path from $(0,0)$ to $(7,12)$ we have a string of 3 $R$s and 6 $U$s.

Therefore, the total number of paths from $(0,0)$ to $(7,12)$ is equal to the total number of strings of 3 $R$s and 6 $U$s, which is equal to: