Set X consists of 10 integers and has the median of 20 and the range

The highlighted part is only applicable when the set is equally spaced (i.e. all terms are in AP)

Ex. take a set

\(1 \quad 5 \quad 9\)

Sum = 15 = 3 * 5

However

\(1 \quad 5 \quad 6\)

Sum = \(12 \neq 3 * 5\)

It should be (average)*(number of terms) = (sum of terms). And only when (average) = (median) (for example when the data set is evenly spaced) it will be the way you've written. Note that (average) = (median) does not necessarily mean that the data set is evenly spaced, for example, in {-3, 2, 2, 7} (average) = (median) = 2 but the set is not evenly spaced.

Set X consists of 10 integers and has the median of 20 and the range of 20. What is the value of the greatest possible integer that can be present in the set?

(A) 32
(B) 37
(C) 40
(D) 43
(E) 50

The range of 20 means that if the smallest term is x, then the largest term is x + 20. In ascending order the data set will be:

{x, ?, ?, ?, ?, ?, ?, ?, ?, x + 20}

The median of 20 means that 20 is the average of two middle numbers. This in turn means that x can never be more than 20 (if, the smallest term, x is more than 20, then the median, the middle term, cannot be 20).

So, the least value of x is 20 and thus the greatest possible integer that can be present in the set is x + 20 = 40. For example, the data set can be:
{20, 20, 20, 20, 20, 20, 20, 20, 20, 40}
{20, 20, 20, 20, 20, 20, 21, 22, 24, 40}
{20, 20, 20, 20, 20, 20, 40, 40, 40, 40}
...

Answer: C.