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CAT 2024 Slot 3 QA Question 18

Type-in-the-answer (no negative marking) · Modern Math · Permutation & Combination · Try it, then check the answer and solution below.

CAT 2024 Slot 3QAModern Math • Permutation & CombinationModerate
The number of all positive integers up to 500 with non-repeating digits is
TITA Answer:
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Answer and solution

Answer: 378

📌 Core Concept
To determine the number of positive integers up to 500 with non-repeating digits, we consider numbers with 1, 2, or 3 digits. For each category, we calculate the count of numbers with unique digits.
🔢 Step-by-Step Solution
1
One-digit numbers (1 to 9):
All one-digit numbers have non-repeating digits.
Count: 99
2
Two-digit numbers (10 to 99):
Tens place: 9 options (1-9)
Units place: 9 options (0-9 excluding the tens digit)
Count: 9×9=819 \times 9 = 81
3
Three-digit numbers (100 to 499):
Hundreds place: 4 options (1-4)
Tens place: 9 options (0-9 excluding the hundreds digit)
Units place: 8 options (0-9 excluding the hundreds and tens digits)
Count: 4×9×8=2884 \times 9 \times 8 = 288
4
Total count:
Sum of one-digit, two-digit, and three-digit counts.
Total: 9+81+288=3789 + 81 + 288 = 378
⚡ 30-Second Shortcut
One-digit: Always 9.
Two-digit: 9 × 9 = 81$.
Three-digit: 4 × 9 × 8 = 288$.
Total: 9 + 81 + 288 = 378$.
🎯 Final Answer
Correct Answer: 378

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