Problem 8 · 1994 AJHSME
Hard
Counting & Probability
digit-sumcounting
For how many three-digit whole numbers does the sum of the digits equal 25?
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Answer: C — 6.
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Hint 1 of 2
Counting up to 25 from scratch is messy — flip it. The biggest a digit sum can be is 9+9+9 = 27, so 25 means you only have to take away 2 from a row of nines.
Still stuck? Show hint 2 →
Hint 2 of 2
Now the question is tiny: how can you remove a total of 2 from three 9s? Find those few digit-patterns, then count how many ways each one can be arranged.
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Approach: count from the maximum 27
- Start at 9,9,9 (sum 27) and remove 2. Either take 2 off one digit → 9, 9, 7; or take 1 off two digits → 9, 8, 8. Those are the only digit-bags that sum to 25 (all digits stay valid, and the lead digit is never 0).
- Count arrangements of each: {9, 9, 7} places the lone 7 in 3 spots → (997), (979), (799). {9, 8, 8} places the lone 9 in 3 spots → (988), (898), (889). That's 3 + 3 = 6 numbers.
- Why 'subtract from the max' wins: when a target is near the ceiling, counting the small shortfall is far easier than building up. The arrangement step — 'how many spots for the odd-one-out' — is the same idea you'll use for permutations with repeats.
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