Question 1
Let $n$ be a positive integer. Let $T$ be the set of points $(x,y)$ in the plane where $x$ and $y$ are non-negative integers with $x+y\lt n$. Each point of $T$ is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ of $T$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.