MathDB

Let

n

be a positive integer. In a coordinate grid, a path from

(0,0)

(2n,2n)

consists of

4n

consecutive unit steps

(1,0)

(0,1)

. Prove that the number of paths that divide the square with vertices

(0,0),(2n,0),(2n,2n),(0,2n)

into 2 regions with even areas is

\frac{{4n \choose 2n} + {2n \choose n}}{2}

2