20. Aroon is asked to choose five integers so that the mode is 2 more than the median and the mean is 2 less than the median. What is the largest possible value of the range of Aroon's five integers?
A 2
B 5
C 12
D 15
$\mathrm{E}$ The largest possible range depends on the integers chosen
20. D Let the mean of Aroon's five integers be $m$. Then the median is $m+2$ and the mode is $m+4$. Hence we can let the five integers in ascending order be $x, y, m+2, m+4, m+4$, where $x$ and $y$ are to be determined. Since the mean of the integers is $m$, their sum is $5 m$.
So $x+y+3 m+10=5 m$. Therefore $x+y=2 m-10$. Now for the range of the integers to be as large as possible, $x$ needs to be as small as possible. In turn, this means that $y$ must be as large as possible. We know that $y$ is an integer which is less than $m+2$, so its maximum value is $m+1$. So the smallest possible value of $x$ is $2 m-10-(m+1)=m-11$.
Hence the largest possible value of the range of the integers is $m+4-(m-11)=15$.