, 2018

Solution:

This condition is equivalent to the first and last step being in different directions, as if you switch directions an odd number of times, you must end in a different direction than you started. If the first step is in the $x$ direction and the last step is in the $y$ direction, it suffices to count the number of paths from $(1,0)$ to $(16,15)$, of which there are $\binom{30}{15}$. Similarly, in the other case, it suffices to count the number of paths from $(0,1)$ to $(15,16)$, of which there are also $\binom{30}{15}$. Therefore the total number of paths is $2 \cdot \binom{30}{15}$.