<p>The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is</p>
Step-by-Step Solution
Key Concept: Integers greater than 6,000 must be either 4-digit numbers with first digit ≥ 6, or any 5-digit number. Organize by counting 4-digit and 5-digit cases separately.
<p><strong>Step 1: Identify valid cases</strong></p><p>For a number > 6,000 using digits {3, 5, 6, 7, 8} without repetition:</p><p>• <strong>4-digit numbers:</strong> First digit must be 6, 7, or 8 (≥ 6)</p><p>• <strong>5-digit numbers:</strong> Any arrangement is automatically > 6,000</p><p><strong>Step 2: Count 4-digit numbers with first digit ∈ {6, 7, 8}</strong></p><p>• Choose first digit: 3 ways (6, 7, or 8)</p><p>• Arrange remaining 4 digits in remaining 3 positions: P(4,3) = 4 × 3 × 2 = 24</p><p>• Total 4-digit numbers: 3 × 24 = 72</p><p><strong>Step 3: Count 5-digit numbers</strong></p><p>• Arrange all 5 digits: P(5,5) = 5! = 120</p><p><strong>Step 4: Total count</strong></p><p>Total = 72 + 120 = 192</p><p>∴ Answer: A</p>
Correct Answer: A