88. In a classroom survey of 28 students, 20 students have at least 1 sibling and 17 students have at least 1 cousin. Of these students, 12 have at least 1 sibling and at least 1 cousin. How many of the students surveyed have no siblings and no cousins?
Answer: C
3 students surveyed have no siblings and no cousins.
To determine how many students have no siblings and no cousins, we can use the principle of inclusion-exclusion. In this case, we find that 12 students have both siblings and cousins, leaving us to calculate the remaining students who do not fall into either category.
A) 1
This option is incorrect because if only 1 student had no siblings and no cousins, it would imply that a large majority of the 28 students had either one or both relatives, which contradicts the calculations based on the given numbers.
B) 2
This option is also incorrect. If only 2 students were without siblings and cousins, it would not align with the exclusion of 12 students who have both, as it would suggest a higher number of students with at least one relative than calculated from the survey data.
C) 3
This option is correct. By calculating the total number of students with at least one sibling or cousin (20 + 17 - 12 = 25), we find that 3 students must be without siblings and cousins, as 28 - 25 equals 3.
D) 4
This option is incorrect. If 4 students had no siblings and no cousins, it would imply that the total number of students with at least one relative would be 24, which does not fit the data provided in the survey results.
Conclusion
The accurate calculation shows that 3 students have neither siblings nor cousins, as derived from the inclusion-exclusion principle. All other options fail to correctly represent the relationships among the surveyed students and the totals provided, confirming that C is the definitive answer.