(ii) Treat each group of trees (hibiscus, jacaranda, oleander) as a single unit or 'super tree'. This gives us 3 'super trees' to arrange. The number of ways to arrange these 3 'super trees' is given by:
\(3!\)
Within each 'super tree', the trees can be arranged internally. The number of ways to arrange the hibiscus trees is \(4!\), the jacaranda trees is \(6!\), and the oleander trees is \(2!\). Therefore, the total number of arrangements is:
\(3! \times 4! \times 6! \times 2! = 207360\)
(iii) First, arrange the 8 non-hibiscus trees (6 jacaranda and 2 oleander) in a line. The number of ways to do this is:
\(8!\)
This creates 9 gaps (including the ends) where the hibiscus trees can be placed. We need to choose 4 out of these 9 gaps to place the hibiscus trees, ensuring no two hibiscus trees are adjacent. The number of ways to choose 4 gaps from 9 is given by:
\(\binom{9}{4}\)
Finally, arrange the 4 hibiscus trees within the chosen gaps. The number of ways to do this is:
\(4!\)
Therefore, the total number of arrangements is:
\(8! \times \binom{9}{4} \times 4! = 121,927,680\)