All About 2018 -- D
Probability
Level
3
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
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The answer is 225.
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1 solution
To reduce the number of distinct values, we need to repeat the values that are already in the list as much as possible. Since the list has a unique mode, which occurs 10 times, the greatest number of occurrences of any other value is 9. Therefore we will have 10 lots of a value, 9 lots of a value, 9 lots of a value, 9 lots of a value.... 9 2 0 1 8 − 1 0 ≈ 2 2 3 . 1 Therefore, we will have 223 groups of 9, and we will also need a 224th group of size 1. Not forgetting the modal value, we will have 2 2 5