Problem 11 · 2014 Math Kangaroo
Hard
Counting & Probability
careful-counting
How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.
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Answer: D — 7
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Hint 1 of 3
Bigger than 10 and smaller than 32 means the number has two digits and starts with 1, 2, or 3.
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Hint 2 of 3
Be neat: write all the numbers that start with 1, then all that start with 2, then those that start with 3.
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Hint 3 of 3
Remember each digit can only be 1, 2, or 3, and don't forget to stop before 32.
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Approach: list the two-digit numbers in order, using only the digits 1, 2, 3
- We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
- Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
- Count the list: 11, 12, 13, 21, 22, 23, 31.
- That makes 7 numbers.
Mark:
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