Multiple choice

A fair six-faced die is rolled $12$ times. The probability that each face turns up twice is equal to

  1. $\dfrac { 12! }{ 6!6!{ 6 }^{ 12 } } $
  2. $\dfrac { { 2 }^{ 12 } }{ { 2 }^{ 6 }{ 6 }^{ 12 } } $
  3. $\dfrac { 12! }{ { 2 }^{ 6 }{ 6 }^{ 12 } } $
  4. $\dfrac { 12! }{ { 6 }^{ 2 }{ 6 }^{ 12 } } $
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C Correct answer
Explanation

The total number of outcomes is 6^12. The number of ways to arrange 12 items where each of the 6 faces appears twice is given by the multinomial coefficient 12! / (2! 2! 2! 2! 2! 2!) = 12! / (2^6). Thus, the probability is 12! / (2^6 * 6^12).

AI explanation

Using the multinomial probability formula, the total number of possible outcomes is 6^12. The number of favorable ways for each face to appear exactly twice is 12! / (2! * 2! * 2! * 2! * 2! * 2!), which simplifies to 12! / 2^6. The required probability is 12! / (2^6 * 6^12).

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