A hard question on inclusion–exclusion for 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
All 43 pupils in a class answer a survey on clubs.
16 are in music, 17 in sports and 15 in art.
4 are in both music and sports, 3 in both sports and art, and 8 in both music and art (each of these includes those in all three).
2 are in all three.
How many pupils are in none of the three clubs?
Show the hints (3)
💡Hint 1: First find how many are in at least one club. Adding the three clubs, who is counted more than once?
💡Hint 2: Pupils in exactly two clubs are counted twice; those in all three, three times. Taking off the three "both" numbers takes the all-three pupils off three times as well.
💡Hint 3: After 16 + 17 + 15 − 4 − 3 − 8, the 2 in all three aren't counted at all; add them back.
Worked solution
Have a go first, then read the solution step by step.
1.The three clubs: 16 + 17 + 15 = 48. That counts pupils in exactly two clubs twice and those in all three three times.
2.Take off the pairs: 48 − 4 − 3 − 8 = 33. Pupils in exactly two clubs are now counted once; all three numbers include the pupils in all three clubs, so they are taken off 3 times and now counted 0 times.
3.Add back all three: 33 + 2 = 35. Now everyone inside the circles is counted exactly once: 35 are in at least one club.
4.In none, outside the circles: 43 − 35 = 8.
⚠️Common mistake: after taking off the pairs, forgetting to add back the 2 in all three.
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