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
Pupils decorate the classroom with 7 paper streamers, each 19 cm long.
They glue them end to end into one long streamer, each two neighbours overlapping by 1 cm.
How long is the finished streamer?
Show the hints (2)
💡Hint 1: Gluing 7 streamers in a line, how many overlaps are there?
💡Hint 2: Each overlap counts 1 cm twice.
Worked solution
Have a go first, then read the solution step by step.
1.Ignoring overlaps: 7 × 19 = 133 cm.
2.There is one overlap between each two neighbours: 7 streamers make 7 − 1 = 6 overlaps, 6 × 1 = 6 cm.
3.The overlaps were counted twice, so take them off once: 133 − 6 = 127 cm.
⚠️Common mistake: thinking there are 7 overlaps and taking off 7 cm.
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