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.