a town council plans to plant 12 trees along the centre of a main road.It has 4 hibiscus trees, 6 Jacaranda trees, 2 oleander trees. How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another
8P8 X 9P4
To solve this problem, we can use the concept of permutations since the order of the trees matters. We'll break it down into two steps:
Step 1: Determine the number of arrangements for the non-hibiscus trees.
Step 2: Determine the number of arrangements for the hibiscus trees.
Step 1:
Out of the total 12 trees, 4 of them are hibiscus trees, so we have 12 - 4 = 8 non-hibiscus trees.
The number of arrangements for the non-hibiscus trees can be calculated as 8!, which is the factorial of 8. It represents the number of ways these 8 trees can be arranged. Therefore, there are 8! = 40,320 arrangements for the non-hibiscus trees.
Step 2:
Since no hibiscus tree can be next to another, we need to arrange the hibiscus trees in between the non-hibiscus trees. We have 4 hibiscus trees, so there are 5 possible positions for them to be placed (1 before the first non-hibiscus, 1 between the first and second non-hibiscus, and so on).
The number of arrangements for the hibiscus trees can be calculated as 5!, which is the factorial of 5. It represents the number of ways these 4 trees can be arranged in 5 positions. Therefore, there are 5! = 120 arrangements for the hibiscus trees.
Step 1 and Step 2 are independent events, so to calculate the total number of arrangements, we multiply the number of arrangements from each step.
Total number of arrangements = Number of arrangements for non-hibiscus trees x Number of arrangements for hibiscus trees
Total number of arrangements = 40,320 x 120
Total number of arrangements = 4,838,400.
Therefore, there are 4,838,400 different arrangements of these 12 trees if no hibiscus tree is next to another.
To determine the number of different arrangements of the trees, we can use the concept of permutations.
1. Start by placing the 6 Jacaranda trees. Since no hibiscus tree should be next to another, we can represent the arrangement of Jacaranda trees as J _ J _ J _ J _ J _, where "_" represents a placeholder for the hibiscus and oleander trees.
2. Now, we need to place the 2 oleander trees in the remaining available spots. We can illustrate this as J O J O J _ J _ J _ J _, where "O" represents the oleander trees.
3. Finally, we need to place the 4 hibiscus trees. We have 8 available spots between and around the Jacaranda and oleander trees. We can visualize this arrangement as: J O J O J _ J _ J _ J _
| _ | _ | _ | _ |
Now, we need to select 4 spots out of the available 8 for the hibiscus trees. This can be calculated using the "combination" formula, which represents the number of ways to choose k items from a set of n items. In this case, we want to select 4 spots (k) out of 8 available (n).
The formula for combinations is: C(n, k) = n! / (k! * (n-k)!)
Using this formula, we plug in the values:
C(8, 4) = 8! / (4! * (8-4)!)
= 8! / (4! * 4!)
= (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1)
= 70
Therefore, there are 70 different arrangements of the 12 trees that can be made if no hibiscus tree is next to another.