How many persons would be reading at most two newspapers?

Concept. In a three-circle numerical Venn diagram every surveyed person lies in exactly one region: a single-circle region (that newspaper only), a two-circle overlap (exactly those two), the central region shared by all three, or the space outside every circle (none). The phrase “at most two” means “two or fewer”, so it covers everyone who reads exactly two, exactly one, or none, and leaves out only the region common to all three circles. Because a single region is left out, the complement rule (at most two) = (survey total) − (all three) is faster than adding the qualifying regions one at a time.

Application. Reading the diagram region by region gives the following counts.

1. Check the reading against the survey size: 3 + 4 + 6 + 12 + 8 + 5 + 2 + 10 = 50, which matches the 50 persons surveyed, so every region has been picked up.

2. Identify the one group that reads more than two newspapers: the central region, holding the 2 persons who read all three.

3. Apply the complement rule: 50 − 2 = 48.

Cross-check. Add the qualifying regions directly instead of subtracting: exactly one newspaper = 3 + 4 + 6 = 13, exactly two newspapers = 12 + 8 + 5 = 25, and no newspaper at all = 10. Their total is 13 + 25 + 10 = 48, which is the same figure, so 48 persons read at most two newspapers.

Contrast. Neighbouring totals answer different questions about the same diagram: 12 + 8 + 5 + 2 = 27 counts readers of at least two newspapers, 12 + 8 + 5 = 25 counts readers of exactly two, and 3 + 4 + 6 + 10 = 23 counts readers of at most one.