Problem 19 · AMC 8 Stretch
Core
Counting & Probability
Arithmetic & Operations
or-process-addand-process-multiply
How many whole numbers less than 1000 can be made if every digit must come from a set of 8 different nonzero digits? (Digits may repeat.)
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Answer: 584 numbers
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Hint 1 of 3
Same idea as the {3,5,6,7,9} problem, but now there are 8 allowed digits and none of them is 0.
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Hint 2 of 3
Cases by length (1, 2, or 3 digits) are an OR process; each case is an AND process with 8 choices per slot.
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Hint 3 of 3
Add \(8 + 8^2 + 8^3\).
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Approach: OR over lengths, AND within each length
- Numbers under 1000 have 1, 2, or 3 digits (separate cases, OR), and each digit slot is freely chosen from the 8 nonzero digits (AND, repeats allowed; no leading-zero worry since 0 isn't allowed).
- Add: \(8 + 8^2 + 8^3 = 8 + 64 + 512 = 584\).
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