Question

The number of ways in which a necklace can be formed by using $5$ identical red beads and $6$ identical black beads is?

• A. $\frac{11!}{6!\times 4!}$
• B. ${}^{11}{P}_6$
• C. $\frac{10!}{2\left(6!\times 5!\right)}$
• D. None of these

Answer 1

The correct option is C

$\frac{10!}{2\left(6!\times 5!\right)}$

Explanation for the correct option(s)

Find the number of ways in which a necklace can be formed by using $5$ identical red beads and $6$ identical black beads

The total beads are $11$

We take circular permutation as necklace is circular.

Condition for circular permutation

The number of circular permutation of $n$ distinct thing is equal to $\left(n-1\right)!$

Hence, the arrangement is equal to $\left(11-1\right)!=10!$

The beads arrangement is identical both clockwise and anti-clockwise, and we have $5$ identical red and $6$ identical black beads.

Hence, the total number of arrangements is equal to $\frac{10!}{2\left(6!\times 5!\right)}$

Therefore, the correct answer is Option C.