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:

Question

An integer is chosen at random from 1 to 20. What is the probability that it is a prime number?

Answer 1

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

The board contains a total of 16 squares.
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 \(k = 2\).
This yields \(C(16, 2) = \frac{16 \times 15}{2 \times 1} = 120\) possible pairs.
Next, we identify pairs of squares that do not share a side, meaning they are not adjacent horizontally or vertically.
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.
The initial count of non-adjacent pairs across all rows and columns is \(3 \times 4 + 3 \times 4 = 24\).
This method may lead to overcounting certain arrangements.
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 \(120 - 24 = 96\)).
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 \(\frac{4}{5}\).

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