Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?
It's a problem from BdMO 2021. I have no idea about how to approach this kind of problems (Cause I'm a beginner). Please help me solve it. And sorry if it's too easy.
Without restrictions the answer would be $\binom{10}5^2=63504$. Now for any invalid pair of paths taken by the two toads, swapping the parts of their trajectories after their first meeting gives a pair of paths where Gamakichi goes to $(7,5)$ and Gamatatsu to $(5,5)$.
Gamatatsu's new path must separate $(0,0)$ from $(7,5)$, so the toads' trajectories under this modified scheme must always cross; the transformation described above can thus always be undone, yielding a bijection between invalid path pairs and path pairs under the modified scheme. The cardinality of the second set is $\binom{12}5\binom85=44352$, so the answer is $63504-44352=19152$.