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Let A, B and C be finite sets with n(A)=10, n(B)=15, n(C)=20, n(A∩B)=8 and n(B∩C)=9. Which of the following can be the value of n(A∪B∪C)?

1. 26
2. 27
3. 28
4. Any one of 26, 27, or 28 may occur

Correct answer: Any one of 26, 27, or 28 may occur

Solution

n(AUBUC) = 45 - 8 - 9 - n(A∩C) + n(A∩B∩C). Since n(A∩C) and n(A∩B∩C) are not fixed, testing feasible (nonnegative-region) values yields totals of 26, 27 and 28. Hence any one of 26, 27 or 28 may occur.

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