2002 IMO Problems/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 $h' \le h,k' \le k$ , 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.

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2002 IMO Problems/Problem 1