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CAT 2023 Slot 1 QA Question 21

Multiple choice (+3 / −1) · Modern Math · Permutation & Combination · Try it, then check the answer and solution below.

CAT 2023 Slot 1QAModern Math • Permutation & CombinationModerate
The number of all natural numbers up to 1000 with non-repeating digits is
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Answer and solution

Answer: A) 738738

📌 Core Concept
To find the number of natural numbers up to 1000 with non-repeating digits, we consider numbers with 1, 2, or 3 digits. For each category, we calculate the count using permutations without repetition.
🔢 Step-by-Step Solution
1
1-digit numbers (1 to 9):
All 1-digit numbers have unique digits.
Count = 9
2
2-digit numbers (10 to 99):
First digit (tens place): 9 options (1-9)
Second digit (units place): 9 options (0-9 excluding the first digit)
Count = 9 × 9 = 81
3
3-digit numbers (100 to 999):
First digit (hundreds place): 9 options (1-9)
Second digit (tens place): 9 options (0-9 excluding the first digit)
Third digit (units place): 8 options (0-9 excluding the first and second digits)
Count = 9 × 9 × 8 = 648
4
Total Count:
Sum of 1-digit, 2-digit, and 3-digit numbers.
Total = 9 + 81 + 648 = 738
⚡ 30-Second Shortcut
Recognize that numbers up to 1000 include 1-digit, 2-digit, and 3-digit numbers.
Use permutations for each category without repeating digits.
Add the counts for each category to get the total.
🎯 Final Answer
Correct Answer: Option A

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