JEE Advanced 2023 — Paper 2 — MATH-IV solution

JEE Advanced2023Paper 2MATH-IV

PARAGRAPH “II”

Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column. Assume that each point $A_{i}$ has an equal chance of being chosen.

Q2. Two distinct points are chosen randomly out of the points $A_{1}, A_{2}, \dots, A_{49}$ . Let $p$ be the probability that they are friends. Then the value of $7 p$ is

Your answer

Solution

Total unordered pairs of points: $(2 49) = \frac{49 \cdot 48}{2} = 1176.$

Number of friendly (adjacent) pairs: each row contains $6$ horizontal adjacent pairs and there are $7$ rows, giving $42$ ; similarly $42$ vertical pairs from $7$ columns. Total friendly pairs $= 84$ .

$p = \frac{84}{1176} = \frac{1}{14} \Rightarrow 7 p = \frac{7}{14} = \frac{1}{2} = 0.5.$

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