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 3 of 3: Take products

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In plain words

Take products

A=\prod_{x=0}^{n-1}a_x=\prod_{y=0}^{n-1}b_y=B

Detailed analysis

Distinct $x$ -coordinates mean one choice from every column, and distinct $y$ -coordinates mean one from every row. The equal multisets therefore give $A=B$ .