Question

The number of ways in which necklace can beformed with $8$ red, $11$ black and $12$ green coloured beeds is

• A. $\frac{31!}{8!11!12!}$
• B. $\frac{31}{2!8!11!12!}$
• C. $\frac{30!}{2!8!11!12!}$
• D. $\frac{31!}{2!8!11!12!}$

Answer 1

Answer: (C)

$\frac{30!}{2!8!11!12!}$

A total of $8+11+12=31$ beads are to be rearranged
$\to$ The no of way of doing this in a circular setup $=(31-1)!=30!$
$\to$ Now, since the order does not matter, as it is a necklace, clockwise and counterclockwise setups are same.
So, no. of ways = $\frac{30!}{2!}$
$\to$ However, because $8$ beads are same $\to$ red
$11$ beads are $\to$ black
$12$ beads are $\to$ green
Total no. of ways $=\frac{30!}{2!8!11!12!}$