The answer is D.

The average of seven distinct integers is 12. That means that the sum of seven distinct integers is 84.

Since the lowest integer in the set is -15, it means that the sum of the biggest six integers is 84-(-15) = 99.

From here, the maximum value can be less, equal, or greater than 84.

Set 1: -15, [ 0, 1, 2, 3, 4, 30, 59 ]
Set 2: -15, [ 0, 1, 2, 3, 4, 5, 84 ]
Set 3: -15, [ -1, 0, 1, 2, 3, 4, 90 ]

Yes, but that makes A correct, not D. Read it again. It says "The maximum possible value of the greatest of these integers".

Can anyone give the solution to this?

My approach:
There are 7 terms, -15 is lowest.
So
(-15+x1+x2+x3+x4+x5+x6)/7 =12
In order to get maximum we set all to zero and then find the average which gives
-15+0+0+0+0+0+x = 84
x = 84+15
x=99

Is this approach correct?

In the given information 1st distinct number is -15 as the problem mentions least
so we can find the sum of the rest 6 distinct numbers, let it be x

-15+x/7 = 12 => -15+x = 84 => x = 99

the sum of remaining 6 distinct number is 99

so, assuming
1st = -15
2nd = 0
3rd = 1
4th = 2
5th = 3
6th = 4
7th = 89

you can plug any number you want which is greater than -15 and sum up to 99, but A will always be the correct one.

Answer is A

Is my assumption correct:
set =-15,15,-14,14,-13,13,84
set2= -15,0,1,2,3,4,78
In this case the answer can be D right?

I did the same, but I think the least can be negative also..

Given that The average (arithmetic mean) of seven distinct integers is 12, and the least of these integers is –15 and we need to find the maximum possible value of the greatest of these integers

============================================================

Theory

‣‣‣ Sum = Mean * Total Number of values.

============================================================

=> Sum = 12 * 7 = 84

Least number is -15, so to find the maximum value of the greatest number we need to make all other numbers as small as possible.

Smallest number is -15, so other integers can be taken close to this
=> Set is -15, -14, -13, -12, -11, -10 and the maximum number = 84 - (-15 -14 -13 -12 -11 -10) = 84 + 75 = 159

Clearly Quantity A(159) > Quantity B(84)

So, Answer will be A.
Hope it helps!

Watch the following video to Learn the Basics of Statistics