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?

Rationale

3 students surveyed have no siblings and no cousins.

To determine how many students have neither siblings nor cousins, we can use the principle of inclusion-exclusion. There are 28 students total, and by subtracting those with at least one sibling or cousin, we find that 3 students do not belong to either group.

A (1) YOUR ANSWER

If only 1 student had no siblings and no cousins, that would imply 27 students have either or both. However, the numbers provided for siblings and cousins, when calculated, indicate that more students must be unaccounted for. Therefore, this option is incorrect.

B (2) YOUR ANSWER

Assuming that 2 students have neither siblings nor cousins would imply that 26 students are involved with at least one of the categories. Yet, the calculations based on the overlaps suggest that this does not hold true, as more students must remain outside both groups. Thus, this option is not valid.

C (3) CORRECT

With 20 students having at least one sibling and 17 having at least one cousin, and 12 of these having both, we calculate:
Total with siblings or cousins = 20 + 17 - 12 = 25.
Thus, the number of students without siblings or cousins = 28 - 25 = 3. This calculation confirms that 3 students indeed have no siblings and no cousins.

D (4) YOUR ANSWER

If 4 students had no siblings and no cousins, then 24 students would have at least one sibling or cousin. This contradicts the calculations, which indicate that only 25 students can be accounted for as having at least one sibling or cousin. Hence, this option is incorrect.

Conclusion

Through the use of inclusion-exclusion, we find that 3 students in the survey have neither siblings nor cousins. This conclusion arises from analyzing the overlap and total counts of students with siblings and cousins, confirming that the correct answer is 3. The logical deductions help clarify the relationships among the surveyed groups, ensuring accurate interpretations of the data provided.

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?

3 students surveyed have no siblings and no cousins.

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