(Q). A rectangle with sides 2n-1 and 2m-1 is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side length is?
Solution:-
total length of horizontal side = 2m – 1 → odd
total length of vertical side = 2n – 1 → odd
A rectangle is formed by two horizontal & two vertical lines.
& total number of hrizontal lines = 2m → Even
& total number of vertical lines = (2n – 1) + 1 = 2n → Even
Since total number of lines are even
Hence half lines will be at odd places & other half will be at even places
Horizontal ⇒ odd lines → m even lines → m
Vertical ⇒ odd lines → n even lines → n
& we know that
even – even = even
odd – odd = even
even – odd = odd
odd – even = odd
Since
& as per question side length of the selected rectangle should be odd.
So as per above formula one side should be odd & the other one should be even to get the length of sides of desired rectangle as odd.
⟹ A vertical line of odd place of the rectangle can be selected in ᵐC₁ = m ways
& other vertical line of odd place can be selected in ᵐC₁ = m ways.
By combining above four we will get a rectangle whose side length is odd
∴ total number of required rectangle is
= n.n.m.m
= n².m² Answer