JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Probability

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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 :

(A) $\frac{4}{5}$ ✓ Correct
(B) $\frac{7}{10}$
(C) $\frac{3}{5}$
(D) $\frac{23}{30}$

Answer: (A)

Solution

Total ways to choose 2 squares out of 16:

(2 16) = 120

Count the pairs that share a side in the

4 \times 4

grid: in each of the 4 rows there are 3 horizontally adjacent pairs, giving

4 \times 3 = 12

; likewise each of the 4 columns gives 3 vertically adjacent pairs, giving another 12. Total pairs sharing a side

= 12 + 12 = 24

.

Probability that the two squares share no side:

1 - \frac{24}{120} = 1 - \frac{1}{5} = \frac{4}{5}

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Single correct option

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