Problem 10 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyconsidering-extreme-cases
Peggy will buy one or more new fish for her tank from 6 identical coral fish, 7 identical angel fish, and 3 identical blue fish. In how many ways can she make her selection?
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Answer: 223 ways
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Hint 1 of 3
The fish of each kind look alike, so only the number of each kind taken matters.
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Hint 2 of 3
For each kind she may take 0 up to all available. Count the choices for each of the three kinds.
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Hint 3 of 3
Coral: 0-6 (7 ways), angel: 0-7 (8 ways), blue: 0-3 (4 ways). Multiply, then remove the empty selection since she buys one or more.
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Approach: AND process by quantity, then subtract the empty case
- Count by quantity of each identical kind (AND process).
- Coral: 0-6 gives 7 ways. Angel: 0-7 gives 8 ways. Blue: 0-3 gives 4 ways. So \(7 \times 8 \times 4 = 224\).
- 'One or more' rules out taking no fish, so subtract that one case: \(224 - 1 = 223\).
Mark:
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