# Pin board rectangles

The image shows a pin board (7x5 in size). On which three distinct* rectangles have been constructed. Using only horizontal and vertical lines for the edges of the rectangles, how many distinct rectangles can be made?

Solution

The answer is 210

The solution to this problem is in practice very cute. A distinct rectangle requires two distinct horizontal nodes and two distinct vertical nodes to describe it, so the number of rectangles corresponds to the number of ways of picking two of the seven horizontal nodes multiplied by the ways of picking two of the five vertical nodes.

$$N_r = ^7\!\!C_2 \times ^5\!\!C_2 = 21 \times 10$$