Probability
Geometric Probability
Grade 12
Question:
<p>The probability that a rectangle picked up from a chessboard has the area 6 cm² where the distance between consecutive parallel lines on the board is 1 cm, is</p>
<p>(a) \(\frac{3}{56}\)</p>
<p>(b) \(\frac{9}{28}\)</p>
<p>(c) \(\frac{9}{56}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Count rectangles with area 6 on an 8×8 chessboard by finding all factor pairs of 6 (1×6 and 2×3), then count valid placements for each dimension pair considering the grid constraints.
<p><strong>Step 1:</strong> An 8×8 chessboard has 9 horizontal and 9 vertical lines. Total rectangles = C(9,2)×C(9,2) = 36×36 = 1296</p><p><strong>Step 2:</strong> Find all rectangles with area 6 cm²: Since distance between lines is 1 cm, dimensions must be factors of 6.</p><p><strong>Step 3:</strong> Factor pairs of 6: (1×6) and (2×3)</p><p><strong>Step 4:</strong> For 1×6 rectangle: Can be placed in (9-1)×(9-6) = 8×3 = 24 ways (horizontal) and (9-6)×(9-1) = 3×8 = 24 ways (vertical) = 48 ways total</p><p><strong>Step 5:</strong> For 2×3 rectangle: Can be placed in (9-2)×(9-3) = 7×6 = 42 ways (horizontal) and (9-3)×(9-2) = 6×7 = 42 ways (vertical) = 84 ways total</p><p><strong>Step 6:</strong> Total favorable rectangles = 48 + 84 = 132</p><p><strong>Step 7:</strong> Probability = 132/1296 = 11/108</p><p>∴ Answer: C</p>
Correct Answer: C