The total number of three-digit numbers, with one digit repeated exactly two times, is
Answer: 243
Solution
We count three-digit numbers where exactly one digit appears twice.
Case 1: The digit 0 is repeated twice. The pattern must be 0X0 or X00 (where X is non-zero). This gives 9 numbers.
Case 2: The digit 0 appears once (not repeated). Choose a non-zero digit to repeat (9 choices), and the positions can be arranged in 2 ways (e.g., AAB patterns where A≠0, B=0). This gives numbers.
Case 3: The digit 0 does not appear. Choose one digit to repeat from {1,2,...,9} (9 choices), choose a different digit from the remaining 8 (8 choices), and arrange them in the 3 positions where the repeated digit can be in any 2 of the 3 positions (3 ways). This gives numbers.
Total: .
Case 1: The digit 0 is repeated twice. The pattern must be 0X0 or X00 (where X is non-zero). This gives 9 numbers.
Case 2: The digit 0 appears once (not repeated). Choose a non-zero digit to repeat (9 choices), and the positions can be arranged in 2 ways (e.g., AAB patterns where A≠0, B=0). This gives numbers.
Case 3: The digit 0 does not appear. Choose one digit to repeat from {1,2,...,9} (9 choices), choose a different digit from the remaining 8 (8 choices), and arrange them in the 3 positions where the repeated digit can be in any 2 of the 3 positions (3 ways). This gives numbers.
Total: .
JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Integer answer