There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed?

The correct option is 36.

To find the total number of rectangles (which includes squares, as every square is a rectangle) formed by a grid of parallel horizontal and vertical lines, we can use combinations.

Step 1: Understand how a rectangle is formed
A rectangle is uniquely formed by selecting 2 horizontal lines to serve as its top and bottom boundaries, and 2 vertical lines to serve as its left and right boundaries.

Step 2: Choose 2 horizontal lines out of 4
The number of ways to choose 2 horizontal lines from 4 available horizontal lines is given by the combination formula:

$\left(\begin{array}{}42\end{array}\right)=\frac{4\times 3}{2\times 1}=6$

Step 3: Choose 2 vertical lines out of 4
Similarly, the number of ways to choose 2 vertical lines from 4 available vertical lines is:

$\left(\begin{array}{}42\end{array}\right)=\frac{4\times 3}{2\times 1}=6$

Step 4: Calculate the total number of rectangles and squares
By the fundamental counting principle, the total number of rectangles and squares formed is the product of the number of ways to choose the horizontal lines and the vertical lines:

$\text{Total}=6\times 6=36$

Therefore, the maximum number of rectangles and squares that can be formed is 36.