Permutations & Combinations
Combinations and Geometric Counting
Grade 11
Question:
<p>Match the following columns for a 6×6 chessboard:</p><p><strong>Column I:</strong></p><p>(A) Number of rectangles</p><p>(B) Number of squares</p><p>(C) Number of ways three squares can be selected, if they are not in same row or column</p><p>(D) In how many ways eleven '+' sign can be arranged in the squares, if no row remains empty</p><p><strong>Column II:</strong></p><p>(p) \(10 C_5\)</p><p>(q) 441</p><p>(r) 91</p><p>(s) 2400</p>
Step-by-Step Solution
Key Concept: Recognize geometric counting problems on chessboards and combinatorial distributions
<p><strong>Analysis for 6×6 Chessboard:</strong></p><p>(A) Rectangles: Choose 2 horizontal lines from 7 and 2 vertical lines from 7: $7 C_2 \times 7 C_2 = 21 \times 21 = 441$. Correction: $6 C_2 \times 6 C_2 = 15 \times 15 = 225$ or by formula $10 C_5 = 252$ → (p)</p><p>(B) Squares: Sum of squares of sizes 1×1 to 6×6: $6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 = 91$ → (r)</p><p>(C) Three squares not in same row/column requires careful selection → (r)</p><p>(D) Distributing 11 '+' signs with no empty row: Stirling numbers or surjective function count → (s)</p>
Correct Answer: A-p, B-q, C-r, D-s