A universe has 8 technology names, 7 financial names and 5 energy names. Six names are drawn at random without replacement. What is the probability that all three sectors are represented?

A universe has 8 technology names, 7 financial names and 5 energy names. Six names are drawn at random without replacement. What is the probability that all three sectors are represented?

Approach: Count the complement with inclusion-exclusion over the events that a given sector is missing, being careful that two sectors cannot always be missing at once, then divide by the total number of six-name subsets.

3115/3876. The binomial coefficients give C(20,6) = 38760 equally likely subsets. Let A, B and C be the events that technology, financials and energy are absent, with |A| = C(12,6) = 924, |B| = C(13,6) = 1716 and |C| = C(15,6) = 5005. For pairs, |A and B| = C(5,6) = 0 because five energy names cannot fill six slots, |A and C| = C(7,6) = 7 and |B and C| = C(8,6) = 28, and all three cannot be absent. Inclusion-exclusion on the complement counting gives 924 + 1716 + 5005 - 0 - 7 - 28 = 7610 bad subsets, so 31150 good ones and 31150/38760 = 3115/3876 = 0.8037. Hypergeometric sampling makes the three sector counts negatively related, which is why the pair terms matter.

Follow-up: What is the probability that exactly two of the three sectors are represented?

Key concepts: inclusion-exclusion, hypergeometric sampling, complement counting, binomial coefficients.