Case 1: n(A) = 25, n(B) = 32

In this case, 25 + 32 = 57 < 60, so it is possible that the 57 people belong in separate groups to leave 3 people ungrouped.

So, the least possible value of x in this case is 0 (a clear separation between A and B).

n(A) = 25, n(B) = 32, n (A n B) = 0, n (A u B)' = 3

The largest possible value of x is of course 25 (all the 25 people in A also belong to 25 of the 32 people present in B). This leaves 28 people ungrouped.

n (A) = 25, n(B) = 32, n (A n B) = 25, n (A u B)' = 28

Case 2: n(A) = 25, n(B) = 32

In this case, 25 + 32 = 57 > 50, so we cannot have a clear separation between A and B. In other words, there cannot be 57 distinct people here.

There is an "excess" of 7 people in this case. So, at least 7 people must overlap between both groups whilst maintaining the fact that n(A) = 25 and n(B) = 32.

If 7 people really overlap, then there are 18 people who are strictly in A and 25 people who are strictly in B. You can figure out in a similar fashion why an overlapping of only 6 people is not possible here (6 + 19 + 26 = 51 > 50).

So, the least possible value of x is 7.

n(A) = 25, n(B) = 32, n (A n B) = 7, n (A u B)' = 0

The largest possible value of x remains at 25, as per Case 1. This time, 18 people are ungrouped.

n(A) = 25, n(B) = 32, n(A n B) = 25, n (a u B)' = 18

Thank you so much for your detailed explanation.