Introduction / Context:
We must deduce certain truths from constraints about gender count and student status in a size-4 group.
Given Data / Assumptions:
- At least two are female ⇒ number of males ≤ 2.
- Exactly (or at least) three are college students (given as “three”).
Concept / Approach:
Use counting logic to test each conclusion for necessity.
Step-by-Step Solution:
• II: If at least two are females, the maximum number of males is 2 ⇒ “at most two males” is necessarily true.• III: With three college students overall and at least two females total, it is possible that none of the females is a student only if all three students are the (at most) two males plus one more male—but we cannot exceed two males. Therefore, at least one of the students must be female ⇒ III is necessarily true.• I: “Two female group members are college students” need not be true (e.g., exactly one female student and two male students also satisfies constraints). Hence I is not necessary.
Verification / Alternative check:
Check extreme configurations to see what is forced vs optional.
Why Other Options Are Wrong:
Options including I claim more than the constraints force.
Common Pitfalls:
Assuming symmetry forces equal splits among genders within the student subset.
Final Answer:
II and III.
Discussion & Comments