Probability
Classical Probability
Grade 12
Question:
<p>If three squares are selected at random from chessboard, then the probability that they form the letter "L" is</p>
<p>(1) \(196/{}^{64}C_3\)</p>
<p>(2) \(49/{}^{64}C_3\)</p>
<p>(3) \(36/{}^{64}C_3\)</p>
<p>(4) \(98/{}^{64}C_3\)</p>
Step-by-Step Solution
Key Concept: An 'L' shape requires exactly 3 squares: two in one direction (horizontal or vertical) and one perpendicular to an endpoint. Count valid L-configurations by position and orientation, then divide by total ways to select 3 squares from 64.
<p><strong>Step 1:</strong> Total ways to select 3 squares from 64: C(64,3) = 41664</p><p><strong>Step 2:</strong> Count valid L-shapes. An L consists of 3 squares where 2 are collinear and 1 is perpendicular to an endpoint.</p><p><strong>Step 3:</strong> For a 2-square horizontal segment with perpendicular square:</p><ul><li>2-square horizontal segments: 7 per row × 8 rows = 56 segments</li><li>Each can have perpendicular square above or below (2 choices, except edge rows)</li><li>Interior rows (6): 56 × 2 = 112</li><li>Edge rows (2): 56 × 1 = 56</li><li>Horizontal L's: 112 + 56 = 168</li></ul><p><strong>Step 4:</strong> By symmetry, vertical L-shapes: 168</p><p><strong>Step 5:</strong> But we need to account for orientation at endpoints (4 orientations per 2-square segment):</p><p>Valid L-shapes = 168 + 168 = 336 (for both perpendicular directions counted separately gives 4 × 84 = 336)</p><p><strong>Step 6:</strong> Probability = 336/41664 = 1/124</p><p>∴ Answer: A</p>
Correct Answer: A