An easy question on inclusion–exclusion for P3–P4, 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 12 pupils.
Each does at least one sport: swimming or running.
9 swim and 7 run.
How many do both?
Show the hints (2)
💡Hint 1: 9 + 7 is more than the class. Why is there extra?
💡Hint 2: Pupils who do both are counted once in each number.
Worked solution
Have a go first, then read the solution step by step.
1.The 9 swimmers are in the left circle, the 7 runners in the right, and those who do both in the overlap. Everyone does at least one, so all 12 are inside the circles.
2.9 + 7 = 16, which is 16 − 12 = 4 more than the class of 12.
3.The extra are those in the overlap, counted twice: 4 do both.
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