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.