**Step 1: Introduction to the Concept** This problem is about **counting the number of possible configurations** ("landscapes") for recycling bin placements at various recycling points in the city of Ploie.sti. The city has \(3n\) recycling points, each assigned a number from 1 to \(3n\). Depending on the recycling point's number, there are different rules for what types of bins or signs can be placed at each point. Our task is to calculate the total number of distinct ways ("landscapes") in which the bins and signs can be arranged, given the rules in parts (a)–(d). --- **Step 2: Analyze Each Part** Let's analyze each part and calculate the number of choices for each group of recycling points. --- **(a) Recycling points 1 to \(n-3\):** At each of these points, **one bin for plastic and one for paper** is placed. There is **no choice** to be made (the bins are fixed). - Number of such points: \(n-3\) - Number of ways: \(1^{n-3} = 1\) --- **(b) Recycling points \(n-2\) to \(n+3\):** At each of these points, **a bin for glass OR a bin for paper** is placed (**choose one of two options**). - Number of such points: \((n+3) - (n-2) + 1 = n+3 - n+2 + 1 = 6\) - Number of ways: \(2^6 = 64\) --- **(c) Recycling points \(n+4\) to \(2n-1\):** At each of these points, **a bin for plastic, OR glass, OR paper** is placed (**choose one of three options**). - Number of such points: \((2n-1) - (n+4) + 1 = 2n-1-n-4+1 = n-4\) - Number of ways: \(3^{n-4}\) --- **(d) Recycling points \(2n\) to \(3n\):** At each of these points, a **sign is placed** (no choice). - Number of such points: \(3n - 2n + 1 = n+1\) - Number of ways: \(1^{n+1} = 1\) --- **Step 3: Multiply the Number of Ways** Since the choices at each group of recycling points are independent, **multiply the number of ways** for each part: \[ \text{Total Number of Landscapes} = 1 \times 2^6 \times 3^{n-4} \times 1 = 64 \times 3^{n-4} \] --- **Step 4: Final Answer** **There are exactly \(64 \times 3^{n-4}\) distinct recycling landscapes possible in Ploie.sti.** --- **Step 5: What We Did in Each Step (Summary in One Sentence Each)** - **Step 1:** Introduced the problem and explained that it is about counting possible configurations. - **Step 2:** Analyzed each group of recycling points and determined the number of choices for each. - **Step 3:** Multiplied the number of choices to get the total number of possible landscapes. - **Step 4:** Gave the concise final answer. - **Step 5:** Summarized the process for each step in one sentence.