<p>Two integers \(x\) and \(y\) are chosen with replacement out of the set \(\{0, 1, 2, 3, \ldots, 10\}\). Then find the probability that \(|x - y| > 5\).</p>
<p>\(\dfrac{30}{121}\)</p>
<p>\(\dfrac{31}{121}\)</p>
<p>\(\dfrac{60}{121}\)</p>
<p>\(\dfrac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Count ordered pairs (x,y) where |x-y| > 5 by considering cases |x-y| = 6, 7, 8, 9, or 10. For each difference d, count pairs systematically: when x-y = d, we need x ≥ d and y = x-d, giving (11-d) pairs; similarly for y-x = d.
<p><strong>Step 1:</strong> Total possible outcomes when choosing x and y with replacement from {0,1,2,...,10} is 11 × 11 = 121.</p><p><strong>Step 2:</strong> Count pairs where |x - y| > 5, which means |x - y| ∈ {6, 7, 8, 9, 10}.</p><p><strong>Step 3:</strong> For x - y = 6: (6,0), (7,1), (8,2), (9,3), (10,4) → 5 pairs</p><p>For x - y = 7: (7,0), (8,1), (9,2), (10,3) → 4 pairs</p><p>For x - y = 8: (8,0), (9,1), (10,2) → 3 pairs</p><p>For x - y = 9: (9,0), (10,1) → 2 pairs</p><p>For x - y = 10: (10,0) → 1 pair</p><p>Subtotal: 5 + 4 + 3 + 2 + 1 = 15 pairs where x > y by more than 5</p><p><strong>Step 4:</strong> By symmetry, pairs where y - x > 5 also give 15 pairs.</p><p><strong>Step 5:</strong> Total favorable outcomes = 15 + 15 = 30</p><p><strong>Step 6:</strong> Probability = 30/121</p><p>∴ Answer: <strong>30/121</strong></p>
Correct Answer: A