The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is

Mean of the six numbers = 15
So, the sum of the numbers = 15 * 6 = 90
As the median is 18, the mean of middle two numbers must be 18 and thus, their sum must be 36.
Also, the mode is a number less than 18. So, the mode must be appearing as the first and the second number of the six given integers, when arranged in ascending order. To maximize the largest integers, the mode must be equal to 1.
Therefore, out of the six integers, two are 1 and 1.
For the middle two numbers whose sum is 36, we cannot have 18 and 18 because then we will have two modes which is inappropriate as per the question.
So, the middle numbers must be 17 and 19.
The fifth integer can be 20.
Maximum possible value of the largest integer = 90 - (1 + 1 + 17 + 19 + 20) = 32
Hence, option D is the correct answer.