On a chessboard with $6$ rows and $9$ columns, the Slow Rook is placed in the bottom-left corner and the Blind King is placed on the top-left corner. Then, $8$ Sleeping Pawns are placed such that no two Sleeping Pawns are in the same column, no Sleeping Pawn shares a row with the Slow Rook or the Blind King, and no Sleeping Pawn is in the rightmost column. The Slow Rook can move vertically or horizontally $1$ tile at a time, the Slow Rook cannot move into any tile containing a Sleeping Pawn, and the Slow Rook takes the shortest path to reach the Blind King. How many ways are there to place the Sleeping Pawns such that the Slow Rook moves exactly $15$ tiles to get to the space containing the Blind King?


Proposed by Harry Chen (Extile)
1. **Understanding the Problem:**
   - We have a $6 \times 9$ chessboard.
   - The Slow Rook starts at the bottom-left corner $(6,1)$.
   - The Blind King is at the top-left corner $(1,1)$.
   - There are $8$ Sleeping Pawns placed such that:
     - No two Sleeping Pawns are in the same column.
     - No Sleeping Pawn shares a row with the Slow Rook or the Blind King.
     - No Sleeping Pawn is in the rightmost column.
   - The Slow Rook moves vertically or horizontally $1$ tile at a time.
   - The Slow Rook cannot move into any tile containing a Sleeping Pawn.
   - The Slow Rook takes the shortest path to reach the Blind King.
   - We need to find the number of ways to place the Sleeping Pawns such that the Slow Rook moves exactly $15$ tiles to get to the Blind King.

2. **Analyzing the Path:**
   - The Slow Rook must move from $(6,1)$ to $(1,1)$.
   - The shortest path involves $5$ vertical moves up and $0$ horizontal moves.
   - Since the Slow Rook must move exactly $15$ tiles, it must make $10$ additional moves horizontally (right and left).

3. **Placing the Pawns:**
   - The Sleeping Pawns must be placed in such a way that they force the Slow Rook to make $10$ additional horizontal moves.
   - The Slow Rook must avoid the rows containing the Sleeping Pawns.

4. **Counting the Sequences:**
   - We need to count the number of valid sequences of moves that result in exactly $15$ moves.
   - The sequence must include $5$ vertical moves up and $10$ horizontal moves (right and left).

5. **Formulating the Problem:**
   - We need to find the number of $15$-move sequences with $5$ vertical moves up and $10$ horizontal moves.
   - The horizontal moves can be either right or left, and the sequence must be such that the Slow Rook avoids the Sleeping Pawns.

6. **Calculating the Number of Sequences:**
   - The number of ways to arrange $5$ vertical moves (up) and $10$ horizontal moves (right and left) is given by the binomial coefficient:
     \[
     \binom{15}{5} = \frac{15!}{5! \cdot 10!}
     \]
   - This counts the number of ways to choose $5$ positions out of $15$ for the vertical moves.

7. **Considering the Constraints:**
   - We need to ensure that the Slow Rook avoids the rows with the Sleeping Pawns.
   - The exact placement of the Sleeping Pawns affects the number of valid sequences.

8. **Final Calculation:**
   - The exact number of valid sequences depends on the placement of the Sleeping Pawns.
   - We need to account for the constraints and calculate the number of valid sequences accordingly.

The final answer is $\boxed{None}$ (since the exact number of valid sequences depends on the placement of the Sleeping Pawns and requires further calculation).