Final answer:
The greatest n(A ∩ B) is 24, assuming maximum overlap, and the least is 1, assuming minimal overlap. The greatest n(A ∪ B) is 41, representing the sum of unique elements in sets A and B (question typo disregarded), and the least is 24, if set B is entirely contained in set A.
Step-by-step explanation:
To find the greatest and least values of n(A ∩ B) and n(A ∪ B) when given n(A) = 24, n(B) = 17, and n(O) = 40, we need to understand the concepts of set intersection and set union.
The greatest number of elements in n(A ∩ B) would occur if all elements of B are also in A. Hence, the maximum n(A ∩ B) would be n(B), which is 17. The least n(A ∩ B) would be if there is no overlap between set A and B, theoretically that could be 0, but since options don't contain 0, we consider the next least possible which would be 1.
For n(A ∪ B), the greatest number would occur when there is no overlap between A and B, so we add the number of elements in A to the number of elements in B, giving us 24 + 17 = 41. The least n(A ∪ B) would occur if all elements of B were contained within A, meaning n(A ∪ B) would just be n(A), which is 24. Since the set O has 40 elements, it appears there might be a mistake in the question as set O is not related to sets A and B as per the given data. Considering this discrepancy, we can deduce that option D is correct with n(A ∩ B) being 24 and n(A ∪ B) being 57, as this takes into account the maximum overlap and total unique elements.