PL-OM 1991 Problem 2

Let $X$ be the set of all lattice points in the plane (points $(x, y)$ with $x, y \in \mathbb{Z}$ ). A path of length $n$ is a chain $(P_0, P_1, ... , P_n)$ of points in $X$ such that $P_{i-1}P_i = 1$ for $i = 1, ... , n$ . Let $F(n)$ be the number of distinct paths beginning in $P_0=(0,0)$ and ending in any point $P_n$ on line $y = 0$ . Prove that $F(n) = \binom{2n}{n}$