<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: Numbers greater than 6,000 must be either 4-digit numbers starting with 6, 7, or 8, OR all possible 5-digit numbers (which automatically exceed 6,000). Systematically count by cases based on the number of digits and leading digit.
<p><strong>Step 1: Identify valid cases</strong></p><p>Numbers > 6,000 can be:</p><ul><li><strong>4-digit numbers:</strong> Must start with 6, 7, or 8 (to exceed 6,000)</li><li><strong>5-digit numbers:</strong> All possible arrangements (all exceed 6,000)</li></ul><p><strong>Step 2: Count 4-digit numbers starting with 6, 7, or 8</strong></p><p>• First digit: 3 choices (6, 7, or 8)</p><p>• Remaining 3 positions: P(4,3) = 4!/(4-3)! = 4 × 3 × 2 = 24</p><p>• Total 4-digit numbers: 3 × 24 = <strong>72</strong></p><p><strong>Step 3: Count all 5-digit numbers</strong></p><p>• Arrange all 5 digits: P(5,5) = 5! = <strong>120</strong></p><p><strong>Step 4: Find total</strong></p><p>Total = 72 + 120 = <strong>192</strong></p><p>∴ Answer: D</p>
Correct Answer: D