Problem 17 · 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 the set {3, 5, 6, 7, 9}? (Digits may repeat.)
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Answer: 155 numbers
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Hint 1 of 3
A number below 1000 has 1, 2, or 3 digits. These are separate cases — an OR (add) process.
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Hint 2 of 3
Within each case, choosing the digits is an AND (multiply) process. Repeats are allowed, so each digit slot has all 5 choices.
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Hint 3 of 3
1-digit: 5 numbers. 2-digit: \(5 \times 5\). 3-digit: \(5 \times 5 \times 5\). Add the three cases.
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Approach: OR over lengths, AND within each length
- A number under 1000 has 1, 2, or 3 digits — three separate cases (OR, so add). In each case every digit slot is freely chosen from the 5 allowed digits (AND, so multiply), and repeats are allowed.
- One digit: 5. Two digits: \(5 \times 5 = 25\). Three digits: \(5 \times 5 \times 5 = 125\).
- Add the cases: \(5 + 25 + 125 = 155\).
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