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. \( \frac{4}{5} \)
• 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

To determine the probability that two randomly selected squares on a 16-square board do not share a side, the problem is analyzed sequentially:

1. The board contains a total of 16 squares.
2. The total number of combinations for selecting two squares from 16 is calculated using the combination formula: \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n = 16\) and \(k = 2\).
3. This yields \(C(16, 2) = \frac{16 \times 15}{2 \times 1} = 120\) possible pairs.
4. Next, we identify pairs of squares that do not share a side, meaning they are not adjacent horizontally or vertically.
5. Within a single row of 4 squares, the number of ways to choose two non-adjacent squares is \(C(4, 2) - 3 = 6 - 3 = 3\). This count is identical for each of the 4 rows and 4 columns.
6. The initial count of non-adjacent pairs across all rows and columns is \(3 \times 4 + 3 \times 4 = 24\).
7. This method may lead to overcounting certain arrangements.
8. Adjusting the total number of pairs by subtracting those that share a side: \(C(16, 2) - 24 = 120 - 24 = 96\). (Note: The original text's calculation \(120 - 24 = 64\) appears to be an error in calculation or logic. The correct number of pairs that *do* share a side is 24. Thus, the number of pairs that *do not* share a side is \(120 - 24 = 96\)).
9. The probability of selecting two squares with no shared side is therefore \(P = \frac{96}{120} = \frac{4}{5}\).

Consequently, the probability that two randomly chosen squares do not share a common side is \(\frac{4}{5}\).