2021 EJC P2 Q8

Mei Li has \(3\) types of gemstones comprising \(6\) Emeralds, \(4\) Sapphires and \(3\) Labradorites. All the gemstones are of different sizes.

1. Find the number of ways Mei Li can arrange all the gemstones in a circle, where the biggest of each type of gemstone are together.[2]
2. Next, Mei Li selects \(9\) gemstones with \(3\) of each type to arrange in a circle. Find the number of ways to do this, such that the gemstones alternate in the following order, clockwise: Emerald, Sapphire, Labradorite.[3]
3. Finally, Mei Li decides to randomly arrange all \(13\) gemstones in a row. Find the number of ways to do this, such that no two Sapphires are next to each other and the Labradorites are all together.[3]

Mei Li selects \(7\) gemstones to be showcased at a conference.

4. Find the number of ways to select the gemstones such that at least one of each type is selected.[3]