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Correct answer: 64
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Let the three-digit number be written as , where:
- is the hundreds digit,
- is the tens digit,
- is the units digit.
We need:
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Also, the number must lie between and .
Since every number from to is a three-digit number, we count valid digit triples with: and then exclude those less than .
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First count all three-digit numbers whose digits sum to .
We solve: with , .
Put: Then:
Number of non-negative integer solutions is:
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Now check digit upper bounds .
Since , at most one variable can exceed .
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If , let . Then: Number of solutions:
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If , let . Then: with . Put : Number of solutions:
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If , similarly number of solutions is:
Hence valid three-digit numbers with digit sum are:
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Now exclude those numbers less than .
Such numbers are from to . We need digit sum .
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If , then: Possible digit pairs are: giving numbers: These are all .
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If , then for numbers , we only have to . Their digit sums are at most , so none has digit sum .
Therefore, exactly such numbers are below .
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Required count:
Therefore, the number of natural numbers between and whose digits sum to is:
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