If a square is divided into \( 4 \times 4 \) squares. If two squares are chosen randomly, then the probability that the squares don't share a common side is:

Question

Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:

Answer 1

The provided square is segmented into \( 4 \times 4 = 16 \) smaller squares.Step 1: Total Combinations for Two SquaresThe total number of ways to select two squares from the 16 is calculated as:\[\binom{16}{2} = \frac{16 \times 15}{2} = 120.\]Step 2: Combinations of Adjacent SquaresTo determine pairs that do not share a side, we first count adjacent pairs.- Horizontal adjacent pairs: \( 4 \times 3 = 12 \).- Vertical adjacent pairs: \( 4 \times 3 = 12 \).- Total adjacent pairs: \( 12 + 12 = 24 \).Step 3: Combinations of Non-Adjacent SquaresThe number of ways to choose two squares without a shared side is:\[120 - 24 = 96.\]Step 4: Probability CalculationThe probability that two selected squares do not share a common side is:\[\text{Probability} = \frac{96}{120} = \frac{4}{5}.\]The final probability is \( \boxed{\frac{4}{5}} \).

If a square is divided into \( 4 \times 4 \) squares. If two squares are chosen randomly, then the probability that the squares don't share a common side is:

The Correct Option is D

Solution and Explanation

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