The city of Ploie¸sti has 3n recycling points where n ≥ 6. The recycling points are
distinguishable and numbered from 1 to 3n. Hoping to get reelected, but having a shortage of
recycling bins Mayor Val decides to place:
(a) At each recycling point numbered from 1 to n − 3 a bin for plastic, one for paper and one
for glass.
(b) At each recycling point numbered from n − 2 to n + 3 a bin for plastic, or one for paper.
(c) At each recycling point numbered from n + 4 to 2n − 1 a bin for plastic, or one for glass
or one for paper.
(d) At each recycling point numbered from 2n to 3n a sign that says “Vote Val (the Best
Recycler) for Mayor”.
How many distinct recycling landscapes can Ploie¸sti possibly have?

To solve this problem, we need to consider the number of distinct recycling landscapes that can be created by the bins placed at the recycling points according to the rules provided.

### Step-by-Step Breakdown:

#### Step 1: Analyzing the Recycling Points
Given:
- The city has \( 3n \) recycling points.
- The recycling points are numbered from 1 to \( 3n \).

#### Step 2: Distribution of Recycling Bins

(a) **Recycling points 1 to \( n-3 \):**  
Each point has a bin for plastic, paper, and glass. Since each recycling point from 1 to \( n-3 \) has a fixed set of 3 bins (plastic, paper, and glass), there is only **1 possible arrangement** per point here. Thus, the total number of distinct landscapes for these points is \( 1 \).

(b) **Recycling points \( n-2 \) to \( n+3 \):**  
Each point has a bin for either plastic **or** paper. Thus, for each of these 6 points, there are 2 choices (plastic or paper). The total number of distinct landscapes for these points is:  
\[
2^6
\]
since there are 6 points, each with 2 possible choices.

(c) **Recycling points \( n+4 \) to \( 2n-1 \):**  
Each point has a bin for plastic **or** glass **or** paper. Thus, for each of these \( n-4 \) points, there are 3 choices. The total number of distinct landscapes for these points is:
\[
3^{n-4}
\]

(d) **Recycling points \( 2n \) to \( 3n \):**  
Each point has only a sign, and no bin is placed. Since the placement is fixed and doesn’t change, there is only **1 possible arrangement** here.

### Step 3: Total Number of Distinct Recycling Landscapes
To get the total number of distinct recycling landscapes, multiply the number of possibilities for each segment:
\[
\text{Total number of distinct recycling landscapes} = 1 \times 2^6 \times 3^{n-4} \times 1 = 64 \times 3^{n-4}
\]

Thus, the number of distinct recycling landscapes that Ploiești can possibly have is:
\[
\boxed{64 \times 3^{n-4}}
\]

This expression gives the total number of distinct recycling landscapes for any \( n \geq 6 \).

---

Do you have any questions or want more details? Here are some related questions to consider:

1. What happens to the total number of landscapes if the number of recycling points changes?
2. How does the formula change if the types of bins available at certain points are restricted differently?
3. Can we generalize this formula to a different set of rules for recycling bins?
4. What if instead of distinct points, we had groups of points sharing the same bin arrangement?
5. How would the outcome change if all points between \( n+4 \) and \( 2n-1 \) had a mandatory plastic bin?

**Tip:** When dealing with combinatorial problems, breaking the problem into distinct cases or steps and then multiplying the outcomes of these cases is a common and effective strategy.

The city of Ploieşti has 3n recycling points where n ≥ 6. The recycling points are distinguishable and numbered from 1 to 3n. Hoping to get reelected, but having a shortage of recycling bins Mayor Val decides to place: (a) At each recycling point numbered from 1 to n − 3 a bin for plastic, one for paper and one for glass. (b) At each recycling point numbered from n − 2 to n + 3 a bin for plastic, or one for paper. (c) At each recycling point numbered from n + 4 to 2n − 1 a bin for plastic, or one for glass or one for paper. (d) At each recycling point numbered from 2n to 3n a sign that says 'Vote Val (the Best Recycler) for Mayor'. How many distinct recycling landscapes can Ploieşti possibly have?

Explore how to calculate the total number of distinct recycling landscapes in Ploieşti, given 3n recycling points with various bin placement rules. This problem involves combinatorics, exponents, and permutations, and is suitable for students in Grades 11-12.

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