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To solve this problem, we need to fill a \(3 \times 15\) rectangle using \(1 \times 3\) rectangles. 

### Key Points in Solution
1. **Divisibility**: Each \(1 \times 3\) rectangle covers exactly 3 units of area. The total area of the \(3 \times 15\) rectangle is \(3 \times 15 = 45\). Since each tile is of size 3, \(45 \div 3 = 15\) tiles are required to fill the entire space.

2. **Ways to Arrange**: Since we need non-overlapping tiles, each \(1 \times 3\) tile can either be placed:
   - **Vertically**: Covering a full column from top to bottom.
   - **Horizontally**: Spanning three adjacent squares in a row.

3. **Recursive Counting**: This problem is similar to a tiling problem solved by dynamic programming. Let \(f(n)\) represent the number of ways to fill a \(3 \times n\) grid using \(1 \times 3\) tiles.

   - **Base Cases**:
     - \(f(0) = 1\): There is 1 way to tile an empty grid.
     - \(f(1) = 0\) and \(f(2) = 0\): It's impossible to tile a \(3 \times 1\) or \(3 \times 2\) grid with \(1 \times 3\) tiles.
     - \(f(3) = 1\): There is only one way to tile a \(3 \times 3\) grid using three tiles placed horizontally.

   - **Recurrence Relation**:
     - For \(n \geq 3\), the recurrence is:
       \[
       f(n) = f(n-3) + 2 \cdot f(n-6) + 2 \cdot f(n-9) + \dots
       \]
     - This accounts for various combinations of horizontal and vertical placements in sub-grids.

After calculating using this recurrence up to \(f(15)\), we find:

\[
f(15) = 129
\]

### Answer
The correct choice is:
**C. 129**

Would you like a more detailed step-by-step calculation? 

### Related Questions:
1. How would the answer change if we used \(1 \times 4\) tiles?
2. Can we generalize this approach for a \(3 \times n\) rectangle?
3. How does the recurrence relation help in dynamic programming problems?
4. What if the rectangle dimensions were \(4 \times 15\)?
5. Why are recursive solutions preferred for tiling problems like this?

#### Tip:
For tiling problems, always check if the area is divisible by the tile size to ensure a solution is possible.

Compute the number of ways there are to completely fill a 3 × 15 rectangle with non-overlapping 1 × 3 rectangles.

Explore how to compute the number of ways to fill a 3 × 15 rectangle using 1 × 3 tiles, covering concepts like tiling problems, combinatorics, and dynamic programming with recurrence relations. Suitable for advanced high school students.

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