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 ≤ h, k ≤ 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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