Verified Solution • 2025–26 Session moderate

QJEEM21Ma75dc: JEE Main 2021 Mathematics - Class 12 JEEM21Ma75dc Solution: Four dice are thrown simultaneously and the numbers shown on…

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 ×\times 2 matrices. The probability that such formed matrix have all different entries and are non-singular, is :

Multiple Choice Options

  • A 45162{{45} \over {162}}
  • B 2181{{21} \over {81}}
  • C 2281{{22} \over {81}}
  • D 43162{{43} \over {162}}

Step-by-Step Verified Solution

A=|abcd|

| A | = ad −- bc

Total case = 64

For non-singular matrix | A | ≠\ne 0 ⇒\Rightarrow ad −- bc ≠\ne 0

⇒\Rightarrow ad ≠\ne bc

And a, b, c, d are all different numbers in the set {1, 2, 3, 4, 5, 6}

Now for ad = bc

(i) 6 ×\times 1 = 2 ×\times 3

⇒\Rightarrow a=6,b=2,c=3,d=1ora=1,b=2,c=3,d=6::} 8 each cases

(ii) 6 ×\times 2 = 3 ×\times 4

⇒\Rightarrow a=6,b=3,c=4,d=2ora=2,b=3,c=4,d=6::} 8 such cases

favourable cases

= 6C4×4!−16^6C_4 \times 4! - 16

required probability

=6C4×4!−1664=43162= {{{^6C_4 \times 4!} - 16} \over {{6^4}}} = {{43} \over {162}}

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