Question 97
An up-right path from $(a, b) \in \mathbb{R}^{2}$ to $(c, d) \in \mathbb{R}^{2}$ is a finite sequence $\left(x_{1}, y_{1}\right), \ldots,\left(x_{k}, y_{k}\right)$ of points in $\mathbb{R}^{2}$ such that $(a, b)=\left(x_{1}, y_{1}\right),(c, d)=\left(x_{k}, y_{k}\right)$, and for each $1 \leq i\lt k$ we have that either $\left(x_{i+1}, y_{i+1}\right)=\left(x_{i}+1, y_{i}\right)$ or $\left(x_{i+1}, y_{i+1}\right)=\left(x_{i}, y_{i}+1\right)$. Two up-right paths are said to intersect if they share any point.
Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0,0)$ to $(4,4)$, $B$ is an up-right path from $(2,0)$ to $(6,4)$, and $A$ and $B$ do not intersect.
Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0,0)$ to $(4,4)$, $B$ is an up-right path from $(2,0)$ to $(6,4)$, and $A$ and $B$ do not intersect.