Problem 9 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyconsidering-extreme-cases
A fruit bowl has 3 apples, 2 oranges, and 4 bananas. Judy will take at least one piece of fruit. How many different fruit combinations can she take? (Pieces of the same fruit look alike, so only how many of each she takes matters.)
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Answer: 59 combinations
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Hint 1 of 4
Since pieces of the same fruit are identical, for each fruit only the quantity matters — like the porter's coins.
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Hint 2 of 4
For each fruit she may take 0 up to all of them. Count the quantity choices for each fruit.
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Hint 3 of 4
Apples: 0-3 (4 ways), oranges: 0-2 (3 ways), bananas: 0-4 (5 ways). Multiply.
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Approach: AND process by quantity, then subtract the empty case
- Identical pieces mean we count by quantity per fruit (AND process).
- Apples: 0-3 gives 4 ways. Oranges: 0-2 gives 3 ways. Bananas: 0-4 gives 5 ways.
- Multiply: \(4 \times 3 \times 5 = 60\).
- This includes taking nothing. Since she takes at least one piece, subtract that case: \(60 - 1 = 59\).
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