Question

If a square is divided into \( 4 \times 4 \) squares. If two squares are chosen randomly, then the probability that the squares don't share a common side is:

• A. \( \frac{5}{16} \)
• B. \( \frac{6}{16} \)
• C. \( \frac{9}{16} \)
• D. \( \frac{4}{5 \)

Answer 1

The Correct Option is D

The given square is divided into \( 4 \times 4 = 16 \) smaller squares. Step 1: Total number of ways to choose two squares The total number of ways to choose two squares from 16 is given by: \[ \binom{16}{2} = \frac{16 \times 15}{2} = 120. \] Step 2: Number of ways in which the chosen squares do not share a common side To calculate this, we first find the number of pairs of squares that share a common side. - There are 4 rows and 4 columns, so the number of horizontal adjacent pairs of squares is \( 4 \times 3 = 12 \). - Similarly, the number of vertical adjacent pairs is also \( 4 \times 3 = 12 \). - Hence, the total number of adjacent pairs is \( 12 + 12 = 24 \). Step 3: Number of ways in which the squares do not share a common side The number of ways to choose two squares that do not share a common side is: \[ 120 - 24 = 96. \] Step 4: Probability The probability that the two chosen squares do not share a common side is: \[ \text{Probability} = \frac{96}{120} = \frac{4}{5}. \] Thus, the probability is \( \boxed{\frac{4}{5}} \).