A probability problem by Clyde Sale
Probability
Level
3
(AMC 2012 question)
Five distinct integers are arranged in order, with the smallest being 5 and the largest being 16. The mean of the five integers is prime, and is also equal to the median. The number of possibilities for the second largest integer is
The answer is 3.
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1 solution
The sample of five distinct integers that has the minimum mean is {5,6,7,8,16}, the one with the maximum mean is {5,16,15,14,13}. The means of these samples are 8.4 and 12.6 respectively. The only prime in this range is 11. Therefore, the mean must be 11. But the problem specifies that the median must equal the mean. Then the sample is of the form:
5, _, 11, _, 16.
Since it's mean is 11 the sample values must total 55 implying that the the total of the unknown values must be 55-(5+11+16)=23. The lower number must be strictly between 5 and 11, hence in the set {6,7,8,9,10}.
23-6=17 is outside the exclusive range, 11 ... 16. 23-7=16 is too 23-8=15 is inside the acceptable range 23-9=14 is inside the acceptable range 23-10=13 is inside the acceptable range.