Problem 16 · 2024 Math Kangaroo
Hard
Counting & Probability
complementary-counting
How many three-digit numbers are there that contain at least one of the digits 1, 2 or 3?
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Answer: E — 606
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Hint 1 of 2
It is easier to count the three-digit numbers that avoid 1, 2 and 3 entirely.
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Hint 2 of 2
Count the bad numbers using only allowed digits in each place, then subtract from 900.
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Approach: complementary counting
- There are 900 three-digit numbers in total.
- Numbers with none of 1,2,3: first digit has 6 choices (4–9), the other two have 7 each (0,4–9).
- That is 6 × 7 × 7 = 294 bad numbers.
- So 900 − 294 = 606 contain at least one of 1, 2 or 3.
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