How many persons would be reading at least two newspapers?

Concept. In a three-circle Venn diagram each number sits in exactly one region, and that region names the precise combination of newspapers those people read; the regions never overlap, so nobody is counted twice. A count of how many people read at least two newspapers is therefore not an inclusion-exclusion sum. It is simply the total of every region lying inside two or more circles, namely the three two-circle overlaps together with the central region shared by all three.

Application.

1. Read the regions off the diagram: only H = 3, only I = 4, only T = 6, and 10 people sit outside all three circles.

2. The two-circle overlaps are H and I = 12, H and T = 8, I and T = 5.

3. The central region, inside H, I and T at the same time, holds 2.

4. Collect every region that lies inside two or more circles: 12 + 8 + 5 + 2.

5. 12 + 8 + 5 = 25, and 25 + 2 = 27, so 27 of the 50 people surveyed read two or more of the three newspapers.

Cross-check.

• The eight regions must exhaust the survey: 3 + 4 + 6 + 12 + 8 + 5 + 2 + 10 = 50, which matches the 50 people surveyed, so no region has been misread.

• Independent route: 50 - 10 outside = 40 people read at least one newspaper, and exactly one newspaper is 3 + 4 + 6 = 13, so at least two is 40 - 13 = 27.

• Watch the wording: exactly two would be 12 + 8 + 5 = 25, which drops the central 2 because those 2 people read three newspapers rather than two, while at least two keeps them.