JEE Main 2025 Jan 23 Shift 2, Mathematics Q10: Introduction To Probability

JEE Main2025Jan 23, Shift 2Mathematics

A board has $16$ squares as shown in the figure :

[ADD IMAGE: 4 by 4 grid of squares (16 unit squares arranged in a 4x4 array)]

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
$\frac{23}{30}$

Solution

Total ways to choose 2 squares out of 16: $(2 16) = 120$ .

Count pairs that share a side. In each row of 4, there are $3$ horizontally adjacent pairs; with $4$ rows that gives $4 \times 3 = 12$ horizontal pairs. By symmetry, $12$ vertical pairs as well. Total adjacent pairs: $24$ .

Probability the two chosen squares share a side: $\frac{24}{120} = \frac{1}{5}$ .

Probability they share no side: $1 - \frac{1}{5} = \frac{4}{5}$ .

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JEE Main 2025 Jan 23 Shift 2, Mathematics Q10: Introduction To Probability

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