JEE Advanced 2023 · Paper 2 · Mathematics Q17

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MathematicsNumerical+3Question 17 of 51

Passage

PARAGRAPH "II"

Consider the $6\times6$ square in the figure. Let $A_1, A_2, \ldots, A_{49}$ 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.

A 6 x 6 square grid of unit cells: 7 horizontal and 7 vertical lines, with a dot at each of the 49 intersection points.

Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$ . Let $p$ be the probability that they are friends. Then the value of $7p$ is _____.

Your answer

Answer and worked solution

Official answer: 0.49

Friend pairs. These are the unit segments of the grid. There are 7 rows with 6 segments each, and 7 columns with 6 each:
$2\times7\times6=84.$

Probability.
$p=\frac{84}{\binom{49}{2}}=\frac{84}{1176}=\frac1{14},\qquad7p=0.5.$

Topics:Classical probability Combinations

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JEE Advanced 2023 · Paper 2, Mathematics question 17