1. **State the problem:** We need to find the number of ways to place two rooks on an 8x8 chessboard so that they do not attack each other. 2. **Important rule:** Rooks attack along the same row or column. To ensure they do not attack each other, the two rooks must be placed in different rows and different columns. 3. **Step 1: Total ways to place two rooks:** There are 64 squares on the board. The number of ways to place two rooks on different squares is $$64 \times 63 = 4032$$ because the first rook can be placed anywhere and the second rook anywhere except the first rook's square. 4. **Step 2: Count placements where rooks attack each other:** Rooks attack if they share the same row or the same column. - Same row: There are 8 rows. In each row, choose 2 squares for the rooks: $$8 \times \binom{8}{2} = 8 \times 28 = 224$$. - Same column: Similarly, $$8 \times \binom{8}{2} = 224$$. 5. **Step 3: Subtract placements where rooks attack each other:** However, placements where rooks share both the same row and column (i.e., same square) are impossible since rooks must be on different squares, so no double counting to correct. 6. **Step 4: Calculate placements where rooks do not attack each other:** $$\text{Non-attacking placements} = 4032 - 224 - 224 = 4032 - 448 = 3584$$ 7. **Step 5: Since the rooks are identical, each pair is counted twice (rook A on square 1 and rook B on square 2, and vice versa). Divide by 2 to avoid double counting:** $$\frac{3584}{2} = 1792$$ **Final answer:** There are **1792** ways to place two rooks on an 8x8 chessboard so that they do not attack each other.