Problem 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<n

. Each point of

T

is coloured red or blue, subject to the following condition: if a point

(x,y)

is red, then all points of

T

with first coordinate at most

x

and second coordinate at most

y

are also red. 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

Step 1 of 3: Count blue points by coordinates

In plain words

Count blue points by coordinates

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 $a_x$ in its column and $b_y$ in its row.