If all positive integers that have at least 1 digit equal to 2 are listed in increasing order, what is the 100th integer on the list?

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Correct answer marked below

261

Correct

262

270

271

279

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Explanation

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Official Explanation

Official

Let us calculate different digit numbers possible:

1) Single digit - Only one number, that is 2.

2) 2-digit numbers:

- Pattern 2A: 2 is tens digit, A can be any of 10 digits (0-9) → 10 numbers.

- Pattern A2: 2 is units digit, A can be any of 9 digits (1-9, since A cannot be 0 for a 2-digit number) → 9 numbers.

- Subtract the overlap (22 counted twice): 10 + 9 - 1 = 18 two-digit numbers.

Total single and two-digit numbers = 1 + 18 = 19.

3) Three-digit numbers:

- From 100 to 199: same count as two-digit numbers (since hundreds digit is fixed as 1, we count numbers with at least one digit 2 among the last two digits) → 19 numbers.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.