An ice-cream store has exactly 6 flavors of ice-cream. Each of 6 friends likes exactly 4 flavors.
Is there guaranteed to be a flavor which at least 4 friends like?
3 answers
Yes. Let's say that there is no flavor which at least 4 friends like. This means that for each flavor, at most 3 friends can like that flavor. This would allow a total of 6 flavors x 3 = 18 preferences, which is less than the number of total preferences (6 friends x 4 preferences = 24). Thus, it is always guaranteed that there will be a flavor at least 4 friends like.
I don't see how this works
Perhaps an easier way to understand it would be this -- assume friends 1, 2, 3 like ice creams A, B, C and friends 4, 5, 6 like ice creams D, E, F. We know that each friend has to like exactly four flavors, so we must assign each of them to one more ice cream. (Right now, three friends like each ice cream.) No matter how you place their choices, the number will exceed three (and thus, at least four) for at least one ice cream.
Therefore, there is always guaranteed to be a flavor that at least 4 friends like. This is an extension of the Pigeonhole Principle -- if there are 8 pigeons and 7 holes, one hole must have at least two pigeons.
I hope this answers your question :)
Therefore, there is always guaranteed to be a flavor that at least 4 friends like. This is an extension of the Pigeonhole Principle -- if there are 8 pigeons and 7 holes, one hole must have at least two pigeons.
I hope this answers your question :)
6 answers