Question

10 couples (20 people) take part in a contest where 10 people are to be selected at random to win a $100 prize.

Part (a) How many possible different combinations of winners are there?
Part (b) What is the probability that one person from each of the 10 couples wins a prize?

Answer 1

Please note ⁿC_r = n! / [(n-r)!*r!].

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(a) We have to choose 10 winners out of 20 people = ²⁰C₁₀ = 184,756 combinations

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(b) Probability = Favourable Outcomes/Total Outcomes

Total Outcomes is as calculated in (a) choosing 10 winners out of 20 people = 184756.

Favourable outcome = choosing 1 person from each couple = ²C₁ = 2 ways

Therefore for 10 couples = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2¹⁰ = 1024 of choosing 10 people, such that there is 1 person from each couple.

Therefore the required probability = 1024/184756 = 0.0055

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