$S$ 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 $h_o ≤ h, k_o ≤ k$, then $(h_o, k_o$
) 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.
I don’t quite understand how the problem work. I suspect that you have choose a point in the plane (x,y)such that $x+y\le n-1$, and then color all the points $(x_o,y_o)$ such that $x_o\le x, y_o\le y $ and then count how many lists that contains $n$ blue points that have different $x$ coordinate and do the same thing again but change it to the $y$ coordinate. Am I right.
Best Answer
Perhaps an illustrative example would help. You need to show that for every $n\ge 1$, and for every valid coloring of $S$, the number of ways to choose $n$ blue elements with distinct first coordinates is equal to the number of ways to choose $n$ blue elements with distinct second coordinates. Below, I give two examples of colorings, one with $n=3$ and the other with $n=4$, together with the list of all the ways to select $n$ blue coordinates. As you can see, the number of ways to make both selections is the same; both ways have two selections in the first coloring, and both ways have four selections for the second coloring. You need to prove this always happens.