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In a group of 30 people, 18 like Tea and 15 like Coffee. How many like both?
3
5
8
12
3
Use the inclusion-exclusion shortcut: Both = (Tea + Coffee) - Total group size.
Use the inclusion-exclusion shortcut: Both = (Tea + Coffee) - Total group size.
Identify Given Data
We are given a total group of 30 people, with 18 liking Tea and 15 liking Coffee.
Apply Formula
Use the formula Total = Tea + Coffee - Both. Plugging in the values gives 30 = 18 + 15 - Both.
Solve for Intersection
Simplify the equation: 30 = 33 - Both, so Both = 33 - 30 = 3.
B: 5 is mathematically incorrect based on the set overlap, C: 8 is incorrect as it overestimates the intersection, D: 12 is incorrect because it violates the total population constraint.
A is correct because the inclusion-exclusion calculation (18 + 15 - 30) precisely yields 3 people who like both.
When the sum of individual categories exceeds the total population, the difference directly gives the number of elements belonging to both categories.
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