Problem 3 · AMC 8 Stretch
Core
Counting & Probability
account-for-all-possibilitiesvisual-representation
Picture a flower with 6 triangular petals around a center. Each petal can be either OPEN or CLOSED, all on its own. How many different open/closed patterns are possible in all?
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Answer: 64 patterns
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Hint 1 of 4
Start small. If there were just 1 petal, how many patterns? If there were 2 petals?
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Hint 2 of 4
Each petal has exactly 2 choices (open or closed), and the petals don't affect each other.
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Hint 3 of 4
Multiply the choices: 2 for petal 1, times 2 for petal 2, and so on for all 6 petals.
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Approach: Multiplication (counting) principle
- Go petal by petal. Petal 1 can be open or closed: 2 choices. For each of those, petal 2 has 2 choices, so 2 petals give \(2 \times 2 = 4\) patterns.
- Three petals give \(2 \times 2 \times 2 = 8\), and so on — each new petal doubles the count.
- For all 6 petals: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = 64\).
- So there are 64 different open/closed patterns. (Fun aside: many of these look the SAME if you rotate or flip the flower; counting only the truly different shapes gives 13, but that uses symmetry ideas beyond this problem.)
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