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

Approach: Draw the three-circle Venn diagram and fill it from the inside out. The 'all three' value seeds every other region, so peel the overlaps off each single total to get the 'exactly one' slivers.

Step 1: Centre region.
All three workshops: \( 8 \). This is shared by every pair count, so subtract it when finding the 'exactly two' bands.

Step 2: Exactly Data.
Start from Data \(=48\), remove the Data-Logic overlap (20) and Data-Verbal overlap (12), then add back the centre (8) since it was removed twice:
\[ 48 - 20 - 12 + 8 = 24. \]
Step 3: Exactly Logic.
\[ 60 - 20 - 15 + 8 = 33. \]
Step 4: Exactly Verbal.
\[ 50 - 15 - 12 + 8 = 31. \]
Step 5: Add the three 'exactly one' regions.
\[ 24 + 33 + 31 = 88. \]

Final Answer: \(\boxed{88}\)

Answer 2

Approach: Compute the "exactly one" count for each workshop separately using the standard exactly-one formula, then add the three together.

For any set \(X\) among three sets, the count in exactly \(X\) alone is \[ |X| - (\text{sum of its two pairwise overlaps}) + (\text{triple overlap}). \]
Exactly Data only: \(48-20-12+8=24\).
Exactly Logic only: \(60-20-15+8=33\).
Exactly Verbal only: \(50-15-12+8=31\).

Adding these three regions (they don't overlap each other by definition): \[ 24+33+31=\boxed{88} \]