<p>The total number of five-digit numbers of different digits in which the digit in the middle is the largest is</p>
Step-by-Step Solution
Key Concept: A five-digit number has the form _ _ _ _ _ where the middle (3rd) position must contain the largest digit. Once we choose which 5 distinct digits to use and place the largest in position 3, we need to arrange the remaining 4 digits in the other 4 positions.
<p><strong>Step 1:</strong> Choose 5 distinct digits from {0,1,2,...,9}. The largest must go in the middle (position 3).</p><p><strong>Step 2:</strong> Case 1 - If 0 is NOT among the 5 chosen digits: Choose 5 from {1,2,...,9}, which is C(9,5) ways. The largest goes in position 3, and arrange remaining 4 in positions 1,2,4,5: 4! ways. Subtotal: C(9,5)×4! = 126×24 = 3024</p><p><strong>Step 3:</strong> Case 2 - If 0 IS among the 5 chosen digits: Choose 4 more from {1,2,...,9}, which is C(9,4) ways. The largest of these 5 goes in position 3. Now arrange the remaining 4 digits (including 0) in positions 1,2,4,5. But position 1 cannot be 0: arrange 3 non-zero digits in position 1 (3 choices), then arrange remaining 3 digits in positions 2,4,5 (3! ways). Subtotal: C(9,4)×3×3! = 126×3×6 = 2268</p><p><strong>Step 4:</strong> Total = 3024 + 2268 = 5292</p><p>∴ Answer: A</p>
Correct Answer: A