A medium question on inclusion–exclusion for P4–P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 qualifying rounds and finals; harder ones have three overlapping groups, or ask who is in exactly two.
The question
A class has 16 pupils.
7 like chocolate ice cream, 6 like strawberry, and 6 like neither.
How many like both?
Show the hints (2)
💡Hint 1: First: how many like at least one?
💡Hint 2: 7 + 6 is more than that. The extra are the pupils counted twice.
Worked solution
Have a go first, then read the solution step by step.
1.The 7 who like chocolate are in the left circle, the 6 who like strawberry in the right, and those who like both in the overlap; the 6 who like neither are outside.
2.Inside the circles (at least one): 16 − 6 = 10.
3.7 + 6 = 13, which is 13 − 10 = 3 more than the 10 inside.
4.The extra are those in the overlap, counted twice: 3 like both.
⚠️Common mistake: taking the class of 16 straight from 7 + 6, forgetting to remove the 6 who like neither.
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