1. ## Necklace

Find number of ways to make a necklace using $\displaystyle 3$ identical diamonds and $\displaystyle 15$identical pearls.

2. ## Re: Necklace

This is a "necklace" so circular so we can take anything to be the "starting point" so lets take one of the pearls as marking the "start". That leaves 17 items to be put on. There are 17! ways to order 17 distinct objects. But there are 3 identical diamonds and there are 3! ways to interchange them that we do not want to count. There are 15 identical pearls so 15! ways to interchange them. That gives a total of 17!/(3! 15!) different necklaces.

3. ## Re: Necklace

$\displaystyle \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{136}{3}$=Number of necklaces

4. ## Re: Necklace

Originally Posted by pankaj
$\displaystyle \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{176}{3}$=Number of necklaces
However, the whole question is much more complicated then that.

You need to study this webpage.
You will see the you must account for reflections and rotations.

If you calculate $\displaystyle \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{176}{3}$ you will see that is not a whole number.