1985 AMC 8 Problem 15

Below is the professionally curated solution for Problem 15 of the 1985 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1985 AMC 8 solutions, or check the answer key.

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Concepts:digits inclusion-exclusion casework

Difficulty rating: 1050

15.

How many whole numbers between $100$ and $400$ contain the digit $2?$

$100$

$120$

$138$

$140$

$148$

Solution:

All $100$ numbers from $200$ to $299$ contain a $2.$

Among $100$ to $199,$ there are $10$ with a $2$ in the tens place and $10$ with a $2$ in the units place, but $122$ is counted twice, giving $10 + 10 - 1 = 19.$ The range $300$ to $399$ similarly contributes $19.$

The total is $100 + 19 + 19 = 138.$

Thus, the correct answer is C .

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