There was a mass fight. At least 70% of participants got an eye injury. At least 75% of participants got an ear injury. At least 80% of participants got a nose injury. At least 85% of participants got an arm injury. At least how many % of participants got eye, ear, nose and arm injury? Find an example where the least found percentage is realized.
Set theory exercise
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combinatorics
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2Well potentially *everybody* got all four injuries. That would be one particular scenario. This is obviously not the scenario minimizing that percentage in the intersection, however. Intuitively, to minimize the intersection of all four, you would want to minimize all the numbers. So to solve the second half, you would assume all the percentages are exact. – 2012-09-12
1 Answers
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The least percentage of people who got all injuries occurs if the properties of not having particular injuries are spread over different people. If the $30\%$ that have no eye injury, the $25\%$ that have no ear injury, the $20\%$ that have no nose injury and the $15\%$ that have no arm injury are all different people, then that still leaves $100\%-(30\%+25\%+20\%+15\%)=10\%$ that have all injuries.