Each senior in a college course wrote a thesis. The lengths

The graph gives us the following information:
No of pages : No of people
0-9 : 1
10 - 19 : 4
20 - 29 : 6
30 - 39 : 7
40 - 49 : 2

There are a total of 20 people (1+4+6+7+2)
The median number of pages will be the mean of the 10th and the 11th terms.
The 10th and 11th seniors lie in the 20-29 pages interval. So the median will lie in the 20 - 29 interval.

The list of the number of pages could look something like this:
List 1: 5, 10, 10, 10, 10, 20, 20, 20, 20, [highlight]20, 20[/highlight], 30, 30, 30, 30, 30, 30, 30, 40, 40
or
List 2: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]29, 29[/highlight], 39, 39, 39, 39, 39, 39, 39, 40, 40
or
List 3: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]27, 29[/highlight], 36, 36, 36, 36, 36, 36, 39, 40, 40
or many many other ways

The highlighted terms are the 10th and the 11th terms.

a) What is the least possible number of seniors whose theses were within six pages of the median length ?

We need to move people away from the 10th and the 11th terms such that they are not within 6 pages of median.
Median can be 20 or 29 or anything in between (e.g. (28+29)/2 = 28.5 or (20+29)/2 = 24.5 etc)
In list 1, median is 20. In list 2, median is 29 (there are only two people within a distance of 6 from 29).
Is it possible to move ALL people 6 pages away from median? No. We have 4 more people in the 20-29 range and even if they all are at 20, the 10th person will be at 27 and 11th will be at 29. So the median will be 28. (see list 3)

So the 10th and the 11th person will always be within 6 pages of the median.

B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
We try to bring the people from all groups as close as possible to the median.
Only people from 10-19, 20-29 and 30-39 can possibly be 6 pages away. Let's bring the median right in the middle so that 10-19 and 30-39 can come within 6 pages of the median.

List: 5, 19, 19, 19, 19, 20, 20, 20, 20, [highlight]20, 29[/highlight], 30, 30, 30, 30, 30, 30, 30, 40, 40
The median here will be (20+29)/2 = 24.5
Therefore we can have 4+6+7 = 17 seniors lie within 6 pages of the median.