Theory question 113
Permutations and Combinations(a) If 18Cr = 18Cr+2, find rC5
(b) In a community, 10% of the people tested positive to the HIV virus. If 6 persons from the community are selected at random, one after the other with replacement, calculate correct to four decimal places, the probability that: (i) exactly 5; (ii) none; (iii) at most 2: tested positive to the virus.
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Model answer
(a) ⁿCᵣ = ⁿCₛ with r ≠ s means r + s = n: r + (r + 2) = 18, r = 8
⁸C₅ = 56
(b) Binomial, n = 6, p = 0.1, q = 0.9
(i) P(5) = 6(0.1)⁵(0.9) = 0.000054 ≈ 0.0001
(ii) P(0) = 0.9⁶ = 0.531441 ≈ 0.5314
(iii) P(≤ 2) = 0.531441 + 6(0.1)(0.9)⁵ + 15(0.1)²(0.9)⁴ = 0.531441 + 0.354294 + 0.098415 = 0.98415 ≈ 0.9842
Explanation
Use ⁿCᵣ = ⁿCₙ₋ᵣ; then the binomial formula ⁶Cᵣ(0.1)ʳ(0.9)⁶⁻ʳ.