Question 54
Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2 n, 2 n)$ consists of $4 n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0)$, $(2 n, 0)$, $(2 n, 2 n)$, $(0,2 n)$ into two regions with even areas is
$$
\frac{\binom{4 n}{2 n}+\binom{2 n}{n}}{2}
$$
$$
\frac{\binom{4 n}{2 n}+\binom{2 n}{n}}{2}
$$