Number of people = 1 + 4 + 6 + 7 + 2 = 20
So, the median is the mean of the 10th and the 11th term.
As it can be seen in the figure, both the 10th and the 11th term are between 20 and 29.
(a) the least possible number of people within 6 pages of median could be 0 when the 10th and the 11th term is 29 (and thus, the median is 29) and the remaining 4 people in the 20-29 range have 20 pages and all the 7 people in 30-39 range have 39 pages. There more possibilities, of course, for the answer to be 0.
However, had the question mentioned that the number of pages are distinct for each senior then it would've been interesting.
Lets take the terms surrounding 10th and 11th terms to be as far away from it as possible to get the least possible numbers around it.
Let the 6th,7th,8th,9th terms be 20,21,22,23. Let the 12th,13th,14th,15th,16th,17th terms be 33,34,35,36,37,38,39.
If the 10th and 11th terms are 28 and 29, the median in 28.5 and there are three numbers - 23, 33, 34 - within 6 pages of the median.
If the median is 27.5 or any other number the value is still 3 or more.
So, 3 should be the answer.
(b) The maximum value can be 6 + 7 = 13 when all the terms of 20-29 and 30-39 range are at or very close to the median.
However, for distinct pages, to find the maximum value of numbers around the median, let the numbers be as close to the 10th and 11th terms as possible. The 6th,7th,8th,9th terms would be 24,25,26,27 and the 10th,11th terms will be 28,29 and 12th,13th,14th,15th,16th,17th,18th terms would be 30, 31,32,33,34,35,36. 23,24,25,26,27 and 30,31,32,33,34 are within 6 pages of the median and then 10 is the answer.
