A [i]domino[/i] is a $2 \times 1$ or $1 \times  2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times  2n$ chessboard so that every $2 \times  2$ square contains at least two uncovered unit squares which lie in the same row or column.
1. **Understanding the Problem:**
   We need to place $n^2$ dominoes on a $2n \times 2n$ chessboard such that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.

2. **Lemma:**
   Partition the $2n \times 2n$ chessboard into $2 \times 2$ squares. Each $2 \times 2$ square must contain exactly one domino.

3. **Proof of Lemma:**
   - Each $2 \times 2$ square has at most 2 covered unit squares.
   - Therefore, the total number of covered squares is at most $2n^2$.
   - Since the dominoes cover exactly $2n^2$ squares, equality must hold.
   - This implies that each $2 \times 2$ square must have exactly 2 covered squares.
   - These two covered squares must be consecutive (either horizontally or vertically) because we cannot have the two opposite corners covered.
   - Hence, each $2 \times 2$ square must contain exactly one domino.

4. **Encoding the Problem:**
   - We can encode the problem as an $n \times n$ grid with arrows representing the direction of the dominoes.
   - The problem is equivalent to ensuring that:
     - If an arrow points in direction $X$, then all squares on the ray from that square in direction $X$ have the same direction.
     - There are no rotations or reflections of the configurations $\uparrow\downarrow$ or $\uparrow\leftarrow$.

5. **Bijection to Paths:**
   - There is a bijection from arrow configurations to paths from opposite corners that travel along the gridlines.
   - The number of such paths is $\binom{2n}{n}^2$.
   - These paths partition the square into 4 regions with natural directions (N, E, S, W), and we fill those with the arrows of that particular direction.

6. **Example:**
   - Consider an $8 \times 8$ grid with arrows filled in as shown in the problem statement.
   - The paths partition the grid into regions, and each region is filled with arrows pointing in the same direction.

7. **Conclusion:**
   - The number of ways to place the dominoes is $\boxed{\binom{2n}{n}^2}$.