Question

A painter draws 64 equal squares of 1 square inch on a square canvas measuring 64 square inches. She chooses two squares (1 square inch each) randomly and then paints them. What is the probability that two painted squares have a common side?

• A. 112/2016
• B. 1/3
• C. 512/10034
• D. 3/97
• E. 7/108

Answer 1

The Correct Option is A

To solve this problem, we must calculate the probability that two randomly chosen squares on a 64-square-inch canvas have a common side.

The canvas consists of 64 squares arranged in an 8x8 grid.

The total number of ways to select any 2 squares out of 64 is given by the combination formula \( \binom{64}{2} \):

\(\binom{64}{2} = \frac{64 \times 63}{2} = 2016\)

Next, we determine how many pairs of neighboring squares share a common side:

• Each row in the grid has 7 horizontal pairs of adjoining squares. Therefore, with 8 rows, there are \(8 \times 7 = 56\) horizontal pairs.
• Each column in the grid also has 7 vertical pairs. Thus, with 8 columns, there are \(8 \times 7 = 56\) vertical pairs.

This results in a total of:

\(56 + 56 = 112\) pairs of squares sharing a common side.

Finally, the probability that two randomly chosen squares have a common side is calculated as:

\(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{112}{2016} = \frac{112}{2016}\)