Question

2. Count all rectangles of positive area formed by segments in a grid of m horizontal lines and n vertical lines.

Answer 1

Answer:

$\frac{m(m+1)}{2}\frac{n(n+1)}{2}$

Step-by-step explanation:

We have the grid has m horizontal lines and n vertical lines

We have to find the number of rectangles

If the grid is 1×1, there is 1 rectangle.

If the grid is 2×1, there will be 2 + 1 = 3 rectangles

If it grid is 3×1, there will be 3 + 2 + 1 = 6 rectangles.

So is there is $n\times 1$ there will be $n+(n-1)+(n-2)+(n-3)...............+1=\frac{n(n+1)}{2}$

If we add one column to $n\times 1$ firstly we will have as many rectangles in the 2nd column as the first,

And then we have that same number of $2\times m$ rectangles.

So for $n\times 2=\frac{3n(n+1)}{2}$ rectanglesAfter solving  this we can say

For $n\times m$ we have $\frac{m(m+1)}{2}\frac{n(n+1)}{2}$ rectangles.