JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Probability

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in

2 \times 2

matrices. The probability that such formed matrices have all different entries and are non-singular, is:

Answer: (C)

Solution

Four throws give

6^{4} = 1296

equally likely matrices, and the entries are all different in

6 \times 5 \times 4 \times 3 = 360

of them. A matrix

(a c b d)

is singular when

a d = b c

, and among four distinct values drawn from

{1, 2, \dots, 6}

the only coincidences of products are

1 \times 6 = 2 \times 3

and

2 \times 6 = 3 \times 4

. For each of the quadruples

{1, 2, 3, 6}

and

{2, 3, 4, 6}

the equal-product pair may sit on the main diagonal or on the other one, and each of the two pairs can be written in

2

orders, giving

2 \times 2 \times 2 = 8

singular matrices per quadruple, so

16

in all. The favourable count is

360 - 16 = 344

and the probability is

\frac{344}{1296} = \frac{43}{162}

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Single correct option