Final answer:
To calculate the maximum number of students who neither like to swim nor sing, we employ the principle of inclusion-exclusion, considering the least overlap between the groups, leading to the conclusion that the maximum number of students who like neither is 10.
Step-by-step explanation:
To determine the maximum number of students who neither like to swim nor sing, we can use the principle of inclusion-exclusion from set theory. According to this principle, we add the number of students who like each activity and then subtract the number who like both activities to avoid double counting. However, the question does not provide the number of students who like both, so we consider the worst-case scenario for the maximum number of students who do not like either activity, which is when the groups of students who like to swim and to sing have the least overlap.
65 students like to swim, 55 students like to sing, and there are 100 students in total. If these two groups had no common members, we would add them up to get 120 students, which is not possible because we only have 100 students. Therefore, the minimum overlap must be 20 students (120-100). If we now consider that 20 is the minimum number of students who like both activities, the remaining number of students who like only one activity or the other is 65 + 55 - 20 = 100, exhausting the entire class.
So the maximum number that likes neither is 100 - 100 = 0 students. However, because there is no 0 option and we must choose the maximum, the next possible minimum number of students that like both swimming and singing is 21, which can give us a maximum number of students who like neither to be 100 - (65 + 55 - 21) = 1. But since this number is not in the answer choices, we proceed to the next minimum possible overlap, which is 22, leading to 100 - (65 + 55 - 22) = 2, and we continue this process until we reach an option that is listed in the answer choices.