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 :

• A. $\frac{3}{}$

Answer 1

$\frac{4}{5}$

Answer 2

Solution:

1. Total pairs of squares:

$$\binom{16}{2} = 120.$$

2. Counting adjacent pairs (sharing a common side):

• Horizontally adjacent: In each of 4 rows, there are 3 pairs.
  Total = $4 \times 3 = 12.$
• Vertically adjacent: In each of 4 columns, there are 3 pairs.
  Total = $4 \times 3 = 12.$
• Overall adjacent pairs: $12 + 12 = 24.$

3. Pairs with no side in common:

$$120 - 24 = 96.$$

4. Probability:

$$\frac{96}{120} = \frac{4}{5}.$$

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Minimal Explanation:
Total ways to select 2 squares is 120. There are 24 pairs that share a side. So, non-adjacent pairs = 120 − 24 = 96. Thus, the probability = 96/120 = 4/5.