<p>How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?</p>
Step-by-Step Solution
Key Concept: A number greater than a million requires at least 7 digits, and the first digit cannot be 0. Count total 7-digit arrangements minus those starting with 0.
<p><strong>Step 1: Identify constraints</strong></p><p>We have digits: 2, 3, 0, 3, 4, 2, 3 (seven digits total)</p><p>Frequency: 2 appears 2 times, 3 appears 3 times, 0 appears 1 time, 4 appears 1 time</p><p>A number greater than a million needs at least 7 digits, so we use all 7 digits.</p><p><strong>Step 2: Calculate total arrangements</strong></p><p>Total arrangements of these 7 digits = 7!/(2!×3!×1!×1!) = 5040/(2×6) = 5040/12 = 420</p><p><strong>Step 3: Subtract invalid arrangements (starting with 0)</strong></p><p>If 0 is first, remaining 6 digits (2,3,3,4,2,3) can be arranged in:</p><p>6!/(2!×3!) = 720/(2×6) = 720/12 = 60 ways</p><p><strong>Step 4: Find final answer</strong></p><p>Valid numbers = Total arrangements - Arrangements starting with 0</p><p>Valid numbers = 420 - 60 = 360</p><p><strong>∴ Answer: 360</strong></p>
Correct Answer: 360