MDM4U Unit 4/5 Test - Full Solutions
Part B: Thinking Solutions
1a. No restrictions:
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We are arranging 9 cards where there are repeated suits but no repeated cards. Total
arrangements = 9! = 362,880
1b. Each suit together:
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- Treat each suit group as a block: [2H], [4S], [3C] → 3! ways to arrange blocks
- Arrange within each group: 2! × 4! × 3!
Total = 3! × 2! × 4! × 3! = 6 × 2 × 24 × 6 = 1,728
2. AMANDA (begins with exactly 2 A’s):
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- Fix 2 A’s at start: _A A _ _ _ _
- Remaining: M, N, D (3 letters) → 3! = 6
3. At least 4 males out of 6 members:
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- Case 1: 4M + 2F → C(9,4) × C(8,2) = 126 × 28 = 3,528
- Case 2: 5M + 1F → C(9,5) × C(8,1) = 126 × 8 = 1,008
- Case 3: 6M → C(9,6) = 84
Total = 3,528 + 1,008 + 84 = 4,620
4. Salmonella test accuracy (Bayes’ Theorem):
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- P(S) = 0.10, P(¬S) = 0.90
- P(Pos|S) = 0.97, P(Pos|¬S) = 0.02
- P(Pos) = (0.97)(0.10) + (0.02)(0.90) = 0.097 + 0.018 = 0.115
- P(S|Pos) = (0.97 × 0.10) / 0.115 = 0.097 / 0.115 ≈ 0.843
So, the probability the test result is accurate ≈ 84.3%