Probability
Multinomial Probability
Grade 12
Question:
<p>A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to</p>
<p>(a) \(\frac{12!}{6^{12} \cdot 2^6}\)</p>
<p>(b) \(\frac{12! \cdot 2^{12}}{6^{12}}\)</p>
<p>(c) \(\frac{12!}{6^{12} \cdot (2!)^6}\)</p>
<p>(d) \(\frac{12!}{(2!)^6}\)</p>
Step-by-Step Solution
Key Concept: Use multinomial probability: divide 12 rolls into 6 groups (one per face) with 2 rolls each. The number of ways is $\frac{12!}{2! \cdot 2! \cdot 2! \cdot 2! \cdot 2! \cdot 2!}$, and probability is this divided by $6^{12}$.
<p>Not provided in source text</p>
Correct Answer: c