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?

The relation between Mean, Median and mode is given by

Mode= 3 Median - 2Mean equation1

According to the given condition

Mode= 2+median equation 2

Putting equation 2 in equation 1

2+median= 3 Median - 2Mean

2 Mean= 3 Median - 2- Median

2 Mean= 2 Median - 2

Mean= Median-1

Mean- Median=1Eq. A

According to the second condition

Median- Mode=-2eq. 3

Putting eq. A in eq. 3

Mean -1 - Mode=-2

Mean- Mode=-1

or

Mode- Mean=1equation 4

Adding equation A and equation4

Mode- Mean=1

Mean- Median=1

Mode+ Median =2 equation5

Again Putting equation2 in equation 5

2+median + Median =2

2 Median =2-2

2 Median =0

Median=0

Mean= 0-1

Mean= 0-1=-1

Mode- Mean=1

Mode-(-1)=1

Mode=0

Putting values in the formula of mean

x bar =∑x/5

(-1)5 = 0+0+ x1+x2+0

-5=  x1+x2

So the highest value is 0.

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