A survey of 300 students found that 180 students indicated they like pizza, and 120 students indicated they like pasta. If 70 students like both pizza and pasta, how many students like neither pizza nor pasta?
- A80
- B70
- C90
- D100
Show answer & explanationAnswer & explanation
Correct answer: B. 70
We can use the Inclusion-Exclusion Principle for two sets: |A ∪ B| = |A| + |B| - |A ∩ B|. Let P be the set of students who like pizza, and T be the set of students who like pasta. |P| = 180, |T| = 120, |P ∩ T| = 70. Number of students who like at least one of the two (pizza or pasta) = |P ∪ T| = 180 + 120 - 70 = 300 - 70 = 230. Total students surveyed = 300. Number of students who like neither = Total students - Number of students who like at least one = 300 - 230 = 70. Therefore, 70 students like neither pizza nor pasta.
Why the other options are wrong
- A. Incorrect. This might arise from a miscalculation in the inclusion-exclusion principle.
- C. Incorrect. This might arise from simply subtracting the 'both' from the 'total' (300-70=230) and then misinterpreting.
- D. Incorrect. This might arise from summing the numbers and subtracting from the total without accounting for overlap (300 - (180+120) = 0, clearly wrong).
Inclusion-Exclusion Principle (Two Sets)
The Inclusion-Exclusion Principle for two sets is a counting technique used to find the number of elements in the union of two sets. It states that the size of the union is the sum of the sizes of the individual sets minus the size of their intersection.
- Formula: |A ∪ B| = |A| + |B| - |A ∩ B|.
- This formula corrects for elements counted twice (those in the intersection).
- Useful for finding 'at least one' or 'neither' scenarios.
- If the sets are disjoint, |A ∩ B| = 0.
Memory trick: Add the parts, subtract the overlap, find the 'either/or' map!