Question

How many necklaces of $12$ beads each can be made from $18$ beads of various colours?

• A. $\frac{119\times 13!}{2}$
• B. ${}^{18}{C}_2$
• C. $18!\times 12!$
• D. None of these

Answer 1

The correct option is A $\frac{119\times 13!}{2}$

In the case of necklace, there is no distinction between the clockwise and anticlockwise arrangements, then the required number of circular permutations
$=\frac{{}^{18}{P}_{12}}{2\times 12}=\frac{18!}{6!\times 24}=\frac{18\times 17\times 16\times 15\times 14\times 13!}{6\times 5\times 4\times 3\times 2\times 1\times 24}=\frac{119\times 13!}{2}$