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 solve the problem of finding the probability that two randomly chosen squares from a 16-square board have no side in common, we begin by analyzing the problem step by step: 

1. Total number of squares on the board is 16.
2. The total number of ways to choose two squares from these 16 squares is calculated using the combination formula: \(C(n, k) = \frac{n!}{k!(n-k)!}\). Here, \(n = 16\) and \(k = 2\).
3. Calculate: \(C(16, 2) = \frac{16 \times 15}{2 \times 1} = 120\) ways.
4. Next, find the number of ways to select two squares such that they have no side in common. Consider that two squares will have no side in common if they are not adjacent horizontally or vertically.
5. Each row has 4 squares. For cells in a single row: \(C(4, 2) - 3 = 6 - 3 = 3\) ways to choose two non-adjacent squares. Similarly, there are 3 non-adjacent ways in each column.
6. Since there are 4 such rows and 4 columns as well, the total number of ways to choose such pairs is: \(3 \times 4 + 3 \times 4 = 24\).
7. However, intersections in the total calculations cannot be counted twice. So, specialized arrangements count should be scrutinized.
8. Let's adjust these based on not being adjacent: \(C(16, 2) - 24 = 120 - 24 = 64\).
9. The probability they have no side in common is \(P = \frac{64}{120} = \frac{4}{5}\).

Thus, the probability that two randomly chosen squares have no side in common is \(\frac{4}{5}\).

Answer 2

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): \[ \frac{96}{120} = \frac{4}{5}. \]