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CAT 2020 PYQsQuantitative Ability. Free, no login required.

Modern Math

past year questions, CAT 2020

Permutation & Combination

past year questions, Modern Math, CAT 2020

Easy

How many integers in the set (100,101,102,…,999)(100,101,102, \ldots, 999) have at least one digit repeated?

Entered answer:

Solution

✅ Correct Answer: 252

We need to find how many numbers from 100 to 999 have at least one digit repeated.

Key insight: Instead of directly counting numbers with repeated digits (which is complicated), we'll use a smart approach - find numbers with NO repeated digits and subtract from the total.

The set contains all three-digit numbers: 100, 101, 102, ..., 999

Total three-digit numbers = 999−100+1=900999 - 100 + 1 = 900

For a three-digit number with format ABC (where A, B, C are the digits):

For the first digit (A):

Can be: 1, 2, 3, 4, 5, 6, 7, 8, 9

Cannot be 0 (since we need a three-digit number)

Choices for A = 9

For the second digit (B):

Can be any digit from 0 to 9, except the digit we used for A

Choices for B = 9

For the third digit (C):

Can be any digit from 0 to 9, except the digits we used for A and B

Choices for C = 8

This ensures all three digits are different, so no repetition occurs.

When making independent choices, we multiply the number of options:

Numbers with NO repeated digits = 9×9×8=6489 \times 9 \times 8 = 648

Using our complement approach:

Numbers with repeated digits = Total numbers - Numbers with NO repeated digits

Numbers with repeated digits = 900−648=252900 - 648 = 252

Answer: 252

Key Takeaway: When "at least one" problems seem complex, try the complement method - it often makes the solution much cleaner and faster!

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