Question

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

• A. $\frac{12!}{6!6!6^{12}}$
• B. $\frac{2^{12}}{2^{6}6^{12}}$
• C. $\frac{12!}{2^{6}6^{12}}$
• D. $\frac{12!}{6^{2}6^{12}}$

Answer 1

Answer: (C)

$\frac{12!}{2^{6}6^{12}}$

Answer 2

Required probability
$={ }^{12} C_{2} \times{ }^{10} C_{2} \times{ }^{8} C_{2} \times{ }^{6} C_{2} \times{ }^{4} C_{2} \times{ }^{2} C_{2} \times\left(\frac{1}{6}\right)^{12}$
$= \frac{12 !}{10 ! \times 2 !} \times \frac{10 !}{8 ! \times 2 !} \times \frac{8 !}{6 ! \times 2 !} \times \frac{6 !}{4 ! \times 2 !} \times \frac{4 !}{2 ! \times 2 !} \times \frac{2 !}{2 ! \times 1 !} \times\left(\frac{1}{6}\right)^{12}$
$=\frac{12 !}{2^{6} \times 6^{12}}$