Question

How many necklaces of $10$ beads each can be made from $20$ beads of different colours?

• A. ${\frac{19!}{10!}}^2$
• B. ${\frac{18!}{10!}}^2$
• C. ${\frac{17!}{10!}}^2$
• D. ${\frac{15!}{10!}}^2$

Answer 1

The correct option is A ${\frac{19!}{10!}}^2$

Here clockwise and anticlockwise arrangements are the same, hence the required circular permutation is
$=\left(\frac{{}^{20}{P}_{10}}{2\times 10}\right)$
$=\frac{20!}{10!\times 2\times 10}$
$=\frac{19!\times 20}{10!\times 20}$
$=\frac{19!}{10!}$.
Now the beads can be arranged in the circular fashion in $(20-1)=19!$ ways.
Hence the required number of ways is
$=\frac{19!}{10!}\times 19!$
$=\frac{19{!}^2}{10!}$.