In a survey involving 500 people, it was found that $46 \%$ of those surveyed like apple, $71 \%$ like vanilla, and $85 \%$ like chocolate ice cream. Are there at least six people among those surveyed who like all three types of ice cream?
Solution. Based on the percentages, 230 people like apple, 355 like vanilla, and 425 like chocolate ice cream. If everyone takes a scoop of the ice cream they like, then a total of $230+355+425=1010$ scoops will be taken. If everyone had taken at most 2 scoops, then a total of at most $500 \cdot 2=1000$ scoops would have been taken. This is 10 less than 1010, which means there will be ten people who eat three scoops. Therefore - slightly exceeding the requirement of the problem - there are ten people among those surveyed who like all three types of ice cream.