Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer. Each point
![(x,y)](/media/m/c/9/1/c91aec4078b932368ded863349deaec5.png)
in the plane, where
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
are non-negative integers with
![x+y<n](/media/m/4/7/0/4701862aae6ca17faedc3b95ce917048.png)
, is coloured red or blue, subject to the following condition: if a point
![(x,y)](/media/m/c/9/1/c91aec4078b932368ded863349deaec5.png)
is red, then so are all points
![(x',y')](/media/m/1/5/8/15890085966de63382dce378911030d1.png)
with
![x'\leq x](/media/m/f/f/d/ffd4a6c512bf761b83607175e703fa5e.png)
and
![y'\leq y](/media/m/8/7/e/87e10f8a17aeb81df3abde81bd60a8fa.png)
. Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be the number of ways to choose
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
blue points with distinct
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
-coordinates, and let
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be the number of ways to choose
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
blue points with distinct
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
-coordinates. Prove that
![A=B](/media/m/6/8/9/689f49b3b57cfd7b358bce6b57563e11.png)
.
%V0
Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y<n$, is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ 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 $A=B$.