Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
In $k$ ways can you place 2 rooks on a chessboard such that they are not in attacking positions, if rooks can attack only in a same row or in a same column? Then $\frac{k}{100}$ is_____.
Step-by-Step Solution
Key Concept: Non-attacking rooks must not share a row or column, so after placing the first rook, exclude its entire row and column (16 squares total) from the remaining 63 positions.
Two rooks attack each other if they share the same row or column. On a standard $8 \times 8$ chessboard, we first place the first rook in any of $64$ squares. The second rook cannot be in the same row (8 squares) or same column (8 squares) as the first rook, leaving $64 - 8 - 8 = 48$ valid positions. Since the order of placing rooks doesn't matter, we divide by $2$: $k = \frac{64 \times 48}{2} = \frac{3072}{2} = 1568$. Therefore, $\frac{k}{100} = \frac{1568}{100} = 15.68$.
Correct Answer: 15.68