Question

A group of 120 students attend at least one of three workshops: Data, Logic, and Verbal.
48 attend Data, 60 attend Logic, 50 attend Verbal.
20 attend both Data & Logic, 15 attend both Logic & Verbal, 12 attend both Data & Verbal, and 8 attend all three.
How many students attend exactly one workshop?

Hint:

When working with three overlapping sets, always use inclusion–exclusion carefully: subtract pairwise intersections, then add the triple intersection back.

Answer 1

Correct Answer: 88

We use the principle of inclusion–exclusion to calculate how many students attend exactly one workshop.
Step 1: Compute the number attending exactly Data.
Students counted in Data = 48. From these, subtract those also in Logic (20), also in Verbal (12), and add back those in all three (8) because they were subtracted twice.
Exactly Data \(= 48 - 20 - 12 + 8 = 24\).
Step 2: Compute the number attending exactly Logic.
Logic count = 60. Subtract those also in Data (20) and Verbal (15), then add back all three (8).
Exactly Logic \(= 60 - 20 - 15 + 8 = 33\).
Step 3: Compute the number attending exactly Verbal.
Verbal count = 50. Subtract those also in Logic (15) and Data (12), then add back all three (8).
Exactly Verbal \(= 50 - 15 - 12 + 8 = 31\).
Step 4: Add all students attending exactly one workshop.
\[ 24 + 33 + 31 = 88 \]
Final Answer: \(\boxed{88}\)