How many rectangles are there with vertices within the following grids? In an m by n grid

Answer 1

Assuming a m x n grid is a m x n tessellation of squares, we can think of this grid as a collection of m+1 parallel horizontal lines and n+1 parallel vertical lines. To construct any rectangle in this grid, we need to choose two horizontal lines and two vertical lines. The number of possible ways to choose two horizontal lines from m + 1 horizontal lines is _m+1C₂ = m(m+1)/2. Similarly, the number of possible ways to choose two vertical lines from n + 1 vertical lines is _n+1C₂ = n(n+1)/2. Thus, the total number of possible rectangles is _m+1C₂*_n+1C₂ = mn(m+1)(n+1)/4 by the Fundamental Counting Principle.