The number of all positive integers up to 500 with non-repeating digits is

Question

UPSC Quizroom · Accepted Answer

Step 1: Count numbers by number of digits

Case 1: 1-digit numbers (1 to 9)

All digits are non-repeating automatically

Count = 9

Case 2: 2-digit numbers (10 to 99)

First digit: 1 to 9 → 9 choices

Second digit: 0 to 9 except first digit → 9 choices

Total = 9 × 9 = 81

Case 3: 3-digit numbers (100 to 499)

We are only counting numbers ≤ 500 → first digit = 1, 2, 3, 4 → 4 choices

Second digit: 0–9 except first digit → 9 choices

Third digit: 0–9 except first two digits → 8 choices

Total = 4 × 9 × 8 = 288

Step 2: Add up all counts

1-digit: 9

2-digit: 81

3-digit: 288

Total = 9 + 81 + 288 = 378

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