Question

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

Answer 1

`(m(m+1))/(2)(n(n+1))/(2)`

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* 1` there will be
`n+(n-1)+(n-2)+(n-3)...............+1=(n(n+1))/(2)`

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

And then we have that same number of
`2* m` rectangles.

So for
`n* 2=(3n(n+1))/(2)` rectanglesAfter solving this we can say

For
`n* m` we have
`(m(m+1))/(2)(n(n+1))/(2)` rectangles.