Number System Question 4 - CAT 2020

Step-by-Step Explanation

The set 100,101,102,…,999 contains all 3-digit numbers.

Total numbers in the set = 999 - 100 + 1 = 900

To find how many numbers have at least one digit repeated, it is much easier to subtract the numbers that have all unique digits from the total pool. Let's count the 3-digit numbers with completely distinct digits:

• Hundreds place: Can be filled by any digit from 1 to 9 (9 choices, as 0 cannot be the first digit).
• Tens place: Can be filled by any digit from 0 to 9 except the one chosen for the hundreds place (9 choices).
• Units place: Can be filled by any digit from 0 to 9 except the two digits already used (8 choices).

Numbers with unique digits = 9 × 9 × 8 = 648

Numbers with repeated digits = 900 - 648 = 252

💡 FundaMakers Tip:

The phrase "at least one" is a clear signal to use the Complement Method. Calculating Total - None is much faster than setting up separate cases for numbers with two repeating digits and numbers with three repeating digits.