Question:
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 :
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 :
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Count matrices with 4 distinct entries from 1-6, then remove those with \(ad = bc\).
Updated On: Nov 4, 2026
- \(\frac{45}{162}\)
- \(\frac{21}{81}\)
- \(\frac{22}{81}\)
- \(\frac{43}{162}\)
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The Correct Option is D
Solution and Explanation
Total outcomes \(= 6^4 = 1296\).
Matrices with all entries different: \(6\cdot 5\cdot 4\cdot 3 = 360\).
Singular means \(ad = bc\). With four distinct numbers from \(\{1,\dots,6\}\), the only possibilities are \(1\cdot 6 = 2\cdot 3\) (set \(\{1,2,3,6\}\)) and \(2\cdot 6 = 3\cdot 4\) (set \(\{2,3,4,6\}\)).
For each set, the pair on the main diagonal can be either of the two pairs (2 ways), and each pair can be ordered in 2 ways: \(2\cdot 2\cdot 2 = 8\) matrices. Total singular \(= 16\).
Favourable \(= 360 - 16 = 344\).
Probability \(= \frac{344}{1296} = \frac{43}{162}\).
Answer: (4) \(\frac{43}{162}\)
Matrices with all entries different: \(6\cdot 5\cdot 4\cdot 3 = 360\).
Singular means \(ad = bc\). With four distinct numbers from \(\{1,\dots,6\}\), the only possibilities are \(1\cdot 6 = 2\cdot 3\) (set \(\{1,2,3,6\}\)) and \(2\cdot 6 = 3\cdot 4\) (set \(\{2,3,4,6\}\)).
For each set, the pair on the main diagonal can be either of the two pairs (2 ways), and each pair can be ordered in 2 ways: \(2\cdot 2\cdot 2 = 8\) matrices. Total singular \(= 16\).
Favourable \(= 360 - 16 = 344\).
Probability \(= \frac{344}{1296} = \frac{43}{162}\).
Answer: (4) \(\frac{43}{162}\)
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