Problem 4 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplylogical-reasoning
A traveler wants to tip a porter using coins from her pocket: 4 pennies, 1 nickel, 1 dime, and 6 quarters. She gives at least one coin. How many different tips are possible? (Pennies look alike, and quarters look alike, so only how many of each you give matters.)
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Answer: 139 tips
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Hint 1 of 4
Now some coins come in identical copies. Saying 'this exact penny or that exact penny' would double-count, because the pennies look the same.
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Hint 2 of 4
Instead, for each kind of coin, just decide HOW MANY to give.
Still stuck? Show hint 3 →
Hint 3 of 4
Pennies: you can give 0, 1, 2, 3, or 4 (that's 5 choices). Nickel: 2 choices. Dime: 2 choices. Quarters: 0 through 6 (that's 7 choices). It's an AND process, so multiply.
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Approach: AND process by quantity, then subtract the empty case
- Because the pennies are identical and the quarters are identical, count by HOW MANY of each kind we give, not which exact coin. That makes it an AND process over the four kinds.
- Pennies: 0, 1, 2, 3, or 4 gives 5 choices. Nickel: give it or not gives 2 choices. Dime: 2 choices. Quarters: 0 through 6 gives 7 choices.
- Multiply: \(5 \times 2 \times 2 \times 7 = 140\).
- This count includes giving nothing. Since she gives at least one coin, subtract that one case: \(140 - 1 = 139\).
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