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Sixty students, thirty-five like tea, thirty like coffee, fifteen like both. How many like neither?
Most candidates start drawing circles and labelling regions, and many of them get there. But the drawing is not the method — it is a way of remembering the method, and it is slower than the method itself.
The idea underneath is called inclusion and exclusion, and it is one sentence long. When you add two groups you have counted everyone in the overlap twice, so subtract the overlap once and the total is right.
That single correction handles two groups. For three groups the same correction is applied twice: subtract each of the three overlaps, then notice that the small region in the middle has now been removed one time too many, and add it back.
Every survey question, every question about students taking subjects or households reading newspapers, is that one idea with different words attached. Learn the formula and the diagram becomes optional — useful for checking, never necessary for solving.
1. In a class of 60 students, 35 like tea, 30 like coffee and 15 like both. How many students like at least one of the two?
2. In the same class of 60, with 35 liking tea, 30 liking coffee and 15 liking both, how many like neither?
3. In the same class of 60, how many students like tea but not coffee?
4. If n(A) = 25, n(B) = 20 and n(A ∪ B) = 38, then n(A ∩ B) is:
5. In a survey of 100 people, 40 read A, 35 read B, 30 read C, 15 read A and B, 12 read B and C, 10 read A and C, and 5 read all three. How many read none of the three?
6. Using the same survey data, how many people read exactly two of the three newspapers?
7. In a group of 50 people, 30 play cricket, 25 play football and 10 play neither. How many play both?
8. Which expression correctly gives the number of elements in A ∪ B?