Question

A board has 16 squares as shown in the figure. Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: \includegraphics[]{10.PNG}

• A. $\frac{4}{5}$
• B. $\frac{7}{10}$
• C. $\frac{3}{5}$
• D. $\frac{23}{30}$

Hint:

When dealing with problems involving probability, first calculate the total number of outcomes, then subtract the unfavorable outcomes to find the number of favorable outcomes. Finally, divide the favorable outcomes by the total to find the probability.

Answer 1

The Correct Option is A

We have a board with 16 squares arranged in a 4x4 grid. We are asked to find the probability that two randomly selected squares do not share a side. Step 1: Total number of ways to choose 2 squares from 16
The total number of ways to choose 2 squares from 16 is given by the combination formula: $$\binom{16}{2} = \frac{16 \times 15}{2} = 120.$$ Step 2: Number of ways in which two squares share a side
To find the number of pairs of squares that share a side, observe that: - Each row of 4 squares has 3 adjacent pairs. - Since there are 4 rows, the total number of horizontal pairs is: $$4 \times 3 = 12.$$ - Each column of 4 squares has 3 adjacent pairs. - Since there are 4 columns, the total number of vertical pairs is: $$4 \times 3 = 12.$$ Thus, the total number of pairs of squares that share a side is: $$12 + 12 = 24.$$ Step 3: Number of ways in which two squares do not share a side
The number of pairs of squares that do not share a side is the total number of pairs minus the number of pairs that share a side: $$120 - 24 = 96.$$ Step 4: Probability that two squares do not share a side
The probability is the ratio of the favorable outcomes (pairs of squares that do not share a side) to the total possible outcomes (all pairs of squares): $$\frac{96}{120} = \frac{4}{5}.$$