Question

Which of the following is an acceptable assignment of students to groups?

• A. John, Kate, and Mark; Luz, Nelson, and Olga; Pat
• B. John, Mark, and Olga; Kate and Luz; Nelson and Pat
• C. Kate, Luz, and Olga; John and Nelson; Mark and Pat
• D. Mark, Nelson, and Pat; John and Kate; Luz and Olga
• E. Nelson, Olga, and Pat; John and Luz; Kate and Mark
• A. John is assigned to the group to which Kate is assigned.
• B. John is assigned to the group to which Nelson is assigned.
• C. Kate is assigned to the group to which Pat is assigned.
• D. Kate is assigned to a group consisting of two students.
• J. Nelson is assigned to the group consisting of three students.
• A. Kate is assigned to the group to which John is assigned.
• B. Kate is assigned to the group to which Mark is assigned.
• C. Luz is assigned to the group to which Olga is assigned.
• D. John is assigned to the group consisting of three students.
• O. Pat is assigned to a group consisting of two students.
• A. John is assigned to the group to which Kate is assigned.
• B. John is assigned to the group to which Nelson is assigned.
• C. Kate is assigned to the group to which Luz is assigned.
• D. Kate is assigned to the group to which Nelson is assigned.
• T. Kate is assigned to the group to which Olga is assigned.
• A. John is assigned to the group to which Mark is assigned.
• B. Luz is assigned to the group to which Kate is assigned.
• C. Nelson is assigned to the group to which Olga is assigned.
• D. John is assigned to the group consisting of three members.
• Y. John and Kate are assigned to different groups from each other.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question requires us to check each of the proposed group assignments against the given rules. The correct answer will be the one that satisfies all conditions.
Step 2: Detailed Explanation:
Let's list the rules and group structure for easy reference:

Structure: One group of 3, two groups of 2. Total 7 students.
Rule 1: John and Luz are in different groups (J \(\neq\) L).
Rule 2: Nelson and Pat are in the same group ({N, P}).
Rule 3: Olga is in the group of three ({O, ?, ?}).
Now, let's test each option:

(A) \{J, K, M\, \{L, N, O\}, \{P\}} - Structure: This has groups of 3, 3, and 1. This violates the required structure of 3, 2, 2. VIOLATION.
(B) \{J, M, O\, \{K, L\}, \{N, P\}} - Structure: Groups of 3, 2, 2. OK. - Rule 3 (Olga): O is in the group of three. OK. - Rule 2 (Nelson/Pat): N and P are together in a group. OK. - Rule 1 (John/Luz): J is in the first group, L is in the second group. They are in different groups. OK. - All rules are satisfied. This is an acceptable assignment.
(C) \{K, L, O\, \{J, N\}, \{M, P\}} - Rule 2 (Nelson/Pat): N and P are in different groups. VIOLATION.
(D) \{M, N, P\, \{J, K\}, \{L, O\}} - Rule 3 (Olga): O is in a group of two. The rule says O must be in the group of three. VIOLATION.
(E) \{N, O, P\, \{J, L\}, \{K, M\}} - Rule 1 (John/Luz): J and L are in the same group. VIOLATION.
Step 3: Final Answer:
Option (B) is the only assignment that satisfies the group structure and all three given conditions.

Answer 2

Step 1: Understanding the Concept:
This is a "must be true" question with a new condition. We need to follow the chain of deductions that results from this new information.
Step 2: Detailed Explanation:
1. New Condition: One of the 2-person groups is \{Luz, Mark\}.
2. Apply Rule 3 (Olga): Olga (O) must be in the 3-person group.
- Group of 3: \{O, ?, ?\}
3. Apply Rule 2 (Nelson/Pat): Nelson (N) and Pat (P) must be together. They can't be in the 3-person group with Olga because that would leave only one spot. So, they must form the other 2-person group.
- Group of 2: \{N, P\}
4. Identify the remaining students: The students placed so far are L, M, O, N, P. The remaining students are John (J) and Kate (K).
5. Place the remaining students: The only available spots are the two remaining places in the 3-person group with Olga.
- Therefore, John and Kate must be in the group with Olga.
- Group of 3: \{O, J, K\}
6. Check for consistency:
- Final groups: \{O, J, K\}, \{L, M\}, \{N, P\}.
- Structure (3, 2, 2): OK.
- Rule 1 (J \(\neq\) L): J is in the first group, L is in the second. OK.
- Rule 2 ({N, P}): OK.
- Rule 3 (O in group of 3): OK.
- The arrangement is valid.
7. Evaluate the options: Based on our deduction that the groups must be \{O, J, K\}, \{L, M\}, \{N, P\}, let's see which statement must be true.
- (A) John is assigned to the group to which Kate is assigned. Yes, they are both in the group with Olga. This must be true.
- (B) John is assigned to the group to which Nelson is assigned. False.
- (C) Kate is assigned to the group to which Pat is assigned. False.
- (D) Kate is assigned to a group of two students. False, she is in the group of three.
- (E) Nelson is assigned to the group of three students. False, he is in a group of two.
Step 3: Final Answer:
The initial condition forces Nelson and Pat into one 2-person group, leaving John and Kate to fill the remaining spots in the 3-person group with Olga. Therefore, John and Kate must be in the same group.

Answer 3

Step 1: Understanding the Concept:
This is a "can be true" (possibility) question with a new condition. We need to find an option that could be true in at least one valid scenario that fits the new condition.
Step 2: Detailed Explanation:
1. New Condition: Nelson (N) is in the same group as Olga (O). 2. Apply Rule 3 (Olga): O must be in the 3-person group. Since N is with O, N is also in the 3-person group. 3. Apply Rule 2 (Nelson/Pat): N must be with Pat (P). Since N is in the 3-person group, P must also be in the 3-person group. 4. Deduce the 3-person group: We now know the 3-person group is composed of Olga, Nelson, and Pat. - Group of 3: \{O, N, P\}. 5. Identify remaining students and groups: The remaining students are John (J), Kate (K), Luz (L), and Mark (M). They must form the two 2-person groups. 6. Apply Rule 1 (J \(\neq\) L): John and Luz cannot be in the same group. Since the remaining students must be split into two pairs, and J and L cannot be a pair, they must be in different groups. 7. Construct the 2-person groups: - Let's place J in one group and L in the other. - Group 2a: \{J, ?\} - Group 2b: \{L, ?\} - The remaining students are K and M. They must fill the remaining spots. There are two possibilities for the pairs: - Possibility 1: \{J, K\} and \{L, M\} - Possibility 2: \{J, M\} and \{L, K\} 8. Evaluate the options based on these possibilities: - (A) Kate is assigned to the group to which John is assigned. This is true in Possibility 1 (\{J, K\}). Since we have found a valid scenario where this is true, this "can be true". - (B) Kate is assigned to the group to which Mark is assigned. This happens in none of our valid scenarios. In Possibility 1, K is with J and M is with L. In Possibility 2, K is with L and M is with J. They are never together. - (C) Luz is assigned to the group to which Olga is assigned. False. Olga is in the group of 3, and Luz is in a group of 2. - (D) John is assigned to the group of three students. False. John must be in a group of two. - (E) Pat is assigned to a group of two students. False. We deduced that Pat must be in the group of three. Step 3: Final Answer:
The initial condition forces the 3-person group to be \{Olga, Nelson, Pat\}. The remaining four students must be split into two pairs, with the constraint that John and Luz are in separate pairs. A possible valid arrangement is the groups \{J, K\} and \{L, M\}. In this scenario, Kate is in the same group as John.

Answer 4

Step 1: Understanding the Concept:
This is a "must be true" question where we deduce group assignments based on given restrictions. Step 2: Detailed Explanation:
1. Initial deductions:
- Nelson (N) must be with Pat (P), forming a 2-person group.
- Olga (O) must be in the 3-person group.
- Mark (M) cannot be with John (J) or P/N. 2. Determine possible groupings:
- Remaining students: John (J), Kate (K), Luz (L), Mark (M).
- The 3-person group must include O and two others.
- The remaining two form the other 2-person group. - Considering all rules, there are two valid scenarios: 1. G1={O,J,K}, G2={N,P}, G3={L,M} 2. G1={O,L,M}, G2={N,P}, G3={J,K} 3. Check the options:
- (A) John with Kate: true in both scenarios. ✅ - (B) John with Nelson: false - (C) Kate with Luz: false - (D) Kate with Nelson: false - (E) Kate with Olga: true in one scenario only Step 3: Final Answer:
The only statement that is always true is that John is in the same group as Kate.
Answer: (A). (Note: The provided key (E) is incorrect.)

Answer 5

Step 1: Understanding the Concept:
This is a "CANNOT be true" (impossibility) question. We are asked to identify a statement that cannot hold under the given conditions. The new condition specifies that John (J) and Olga (O) are in different groups. Step 2: Detailed Explanation:
1. Apply Rule 3 (Olga): Olga must be in the group of three members. This is fixed for all possible arrangements. 2. Apply the new condition: Since John is in a different group from Olga, he cannot be in the 3-person group. This forces John into one of the two 2-person groups. 3. Analyze the options: - (D) states that "John is assigned to the group consisting of three members." - Based on our deduction, this is impossible because John cannot be in Olga's 3-person group. Therefore, (D) is a direct contradiction and cannot be true. 4. Check other options: - For example, (C) states "Nelson is assigned to the group to which Olga is assigned." - We can construct a valid scenario where this is true: if Nelson is with Olga in the 3-person group and Pat joins them, the remaining students can form the two 2-person groups while obeying all other rules. - Therefore, (C) is possible and not the correct answer for "CANNOT be true." 5. Conclusion: Among all the options, only (D) directly contradicts the rules and new condition. Step 3: Final Answer:
Given that Olga is in the 3-person group and John is in a different group, John cannot be in the 3-person group. Therefore, the statement "John is assigned to the group consisting of three members" is impossible. Answer: (D). (Note: The provided answer key listing (C) appears to be incorrect.)