International Mathematical Olympiad 2002 Problem 1

$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h'\leq h,k'\leq k$, then $(h',k')$ is also red. A type 1 subset of $S$ has $n$ blue elements with different first member and a type 2 subset of $S$ has $n$ blue elements with different second member. Show that there are the same number of type 1 and type 2 subsets.