MathLabs

Problem 1

Let nn be a positive integer. Let TT be the set of points (x,y)(x,y) in the plane where xx and yy are non-negative integers with x+y<nx+y<n. Each point of TT is coloured red or blue, subject to the following condition: if a point (x,y)(x,y) is red, then all points of TT with first coordinate at most xx and second coordinate at most yy are also red. Let AA be the number of ways to choose nn blue points with distinct xx-coordinates, and let BB be the number of ways to choose nn blue points with distinct yy-coordinates. Prove that A=BA=B.
Step 1 of 3: Count blue points by coordinates
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In plain words

Count blue points by coordinates

ax=#{(x,y)∈T: (x,y) is blue}andby=#{(x,y)∈T: (x,y) is blue}a_x=\#\{(x,y)\in T:\ (x,y)\text{ is blue}\} \quad\text{and}\quad b_y=\#\{(x,y)\in T:\ (x,y)\text{ is blue}\}
Detailed analysis

For each coordinate define the blue counts axa_x in its column and byb_y in its row.