Question

Which of the following lists the patients in an order in which their scheduled appointments can occur, from appointment 1 through appointment 5 ?

• A. J, K, N, L, M
• B. J, M, N, K, L
• C. K, J, N, M, L
• D. L, J, K, N, M
• E. L, J, N, M, K
• A. K is scheduled for appointment 3.
• B. K is scheduled for appointment 4.
• C. L is scheduled for appointment 4.
• D. L is scheduled for appointment 5.
• J. M is scheduled for appointment 1.
• A. J is scheduled for appointment 1.
• B. J is scheduled for appointment 2.
• C. J is scheduled for appointment 4.
• D. K is scheduled for appointment 4.
• O. N is scheduled for appointment 5.
• A. K
• B. J, K
• C. J, M
• D. J, K, L
• T. K, L, M
• A. J's appointment is appointment 1.
• B. N's appointment is appointment 1.
• C. J's appointment is earlier than K's appointment.
• D. K's appointment is earlier than L's appointment.
• Y. N's appointment is earlier than L's appointment.
• A. J is scheduled for appointment 4.
• B. K is scheduled for appointment 5.
• C. L is scheduled for appointment 1.
• D. M is scheduled for appointment 4.
• ^. N is scheduled for appointment 2.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question asks for a valid, complete schedule. We can test each option against the rules and deductions we've established.
Step 2: Key Formula or Approach:
Use the strongest deductions first to eliminate invalid options. We know N must be in slot 3 and M must be in slot 4 or 5.
Step 3: Detailed Explanation:
- (A) J, K, N, L, M: N is in slot 3. M is in slot 5. L is in slot 4. L prefers 1 and only accepts 5 otherwise. Slot 4 is unacceptable for L. Invalid.
- (B) J, M, N, K, L: N is in slot 3. M is in slot 2. This violates the rule that M must be in 4 or 5. Invalid.
- (C) K, J, N, M, L: N is in slot 3. M is in slot 4. L is in slot 5. This is acceptable for M and L. But K is in slot 1. K accepts any appointment except 1. Slot 1 is unacceptable for K. Invalid.
- (D) L, J, K, N, M: N is in slot 4. This violates the rule that N must be in 3. Invalid.
- (E) L, J, N, M, K: N is in slot 3. M is in slot 4. K is in slot 5. L is in slot 1. J is in slot 2. This arrangement respects all placement rules. Let's check preferences: L=1 (pref), J=2 (pref), N=3 (pref), M=4 (pref), K=5 (not pref). Four patients (L, J, N, M) get their preference. This is a valid schedule (Scenario B). Valid.
Step 4: Final Answer:
The only sequence that satisfies all the conditions is (E).

Answer 2

Step 1: Understanding the Concept:
We are given a new condition (J = 2) and asked to find a possible outcome.

Step 2: Key Formula or Approach:
Start with the new condition and combine it with the initial deductions to determine the possible arrangements.

Step 3: Detailed Explanation:
1. New condition: J is in slot 2. This is one of J's preferred slots.
2. Permanent rule: N is in slot 3.
3. So, the schedule is partially: _, J, N, _, _. Slots 1, 4, 5 are open.
4. We need to place L, K, and M. M must be in 4 or 5.
5. Four preferences must be met. We already have J (in 2), N (in 3), and M (in 4 or 5) getting their preferences. That's three. We need at least one more from L or K.
6. K prefers slot 2, which is now taken by J. So K will not get its preference.
7. Therefore, L must get its preference to meet the "at least 4" rule. L's preference is slot 1. So, L must be in slot 1.
8. The schedule is now: L, J, N, _, _. Slots 4 and 5 are for K and M.
9. This corresponds to Scenario B. The final arrangement must be L(1), J(2), N(3), with K and M in slots 4 and 5 in either order.
10. Now, let's check the options:
  - (A) K is scheduled for appointment 3. False, N is in 3.
  - (B) K is scheduled for appointment 4. True, this is possible if M is in 5.
  - (C) L is scheduled for appointment 4. False, L must be in 1.
  - (D) L is scheduled for appointment 5. False, L must be in 1.
  - (E) M is scheduled for appointment 1. False, L must be in 1.

Step 4: Final Answer:
Given J = 2, the only possible arrangement is L = 1, J = 2, N = 3, and {K, M} in {4, 5}. Therefore, K can be in appointment 4.

Answer 3

Step 1: Understanding the Concept:
We are given a new condition (L=5) and asked what statement must logically follow.
Step 2: Key Formula or Approach:
Apply the new condition and see how it constrains the other variables based on the original rules.
Step 3: Detailed Explanation:
1. New condition: L is in slot 5. L's preference is 1, but L accepts 5. This means L is the one patient who is not getting their preference.
2. Because exactly 4 patients must get their preference, the other four patients (J, K, N, M) must be scheduled in their preferred slots.
3. N prefers 3 \(\rightarrow\) N must be in slot 3.
4. K prefers 2 \(\rightarrow\) K must be in slot 2.
5. J prefers 1 or 2. Since slot 2 is taken by K, J must be in slot 1.
6. M prefers 4 or 5. Since slot 5 is taken by L, M must be in slot 4.
7. This determines the entire schedule completely: J(1), K(2), N(3), M(4), L(5). This is our Scenario C.
8. Now we check the options to see what "must be true":
- (A) J is scheduled for appointment 1. True, as deduced above.
- (B) J is scheduled for appointment 2. False.
- (C) J is scheduled for appointment 4. False.
- (D) K is scheduled for appointment 4. False.
- (E) N is scheduled for appointment 5. False.
Step 4: Final Answer:
If L is in appointment 5, the entire schedule is fixed, and J must be in appointment 1.

Answer 4

Step 1: Understanding the Concept:
This question asks for an exhaustive list of all possibilities for a specific slot (appointment 2).
Step 2: Key Formula or Approach:
We need to test each patient to see if they can be validly placed in slot 2. We can use our established scenarios or check from scratch.
Step 3: Detailed Explanation:
- Can J be in 2? Yes. As shown in question 4, if J=2, a valid schedule is L(1), J(2), N(3), K(4), M(5). So, J is possible.
- Can K be in 2? Yes. This is K's preference. In Scenario A, we have L(1), K(2), N(3), J(4), M(5). This is a valid schedule. So, K is possible.
- Can L be in 2? No. L prefers 1 and only accepts 5 otherwise. Slot 2 is an unacceptable appointment for L.
- Can M be in 2? No. M accepts only appointments 4 or 5. Slot 2 is unacceptable for M.
- Can N be in 2? No. N accepts only appointment 3. Slot 2 is unacceptable for N.
Therefore, the only patients who can be scheduled for appointment 2 are J and K. The complete and accurate list is {J, K}.
Step 4: Final Answer:
After testing each patient, only J and K can be placed in appointment 2 without violating any rules.

Answer 5

Step 1: Understanding the Concept:
Given the condition M=5, we need to find which of the statements can be true by finding at least one valid schedule that includes it.
Step 2: Key Formula or Approach:
Fix M=5 and N=3. Then, determine the possible arrangements for J, K, and L in the remaining slots (1, 2, 4) that satisfy the "at least 4 preferences" rule.
Step 3: Detailed Explanation:
1. We are given M=5 (preference) and we know N=3 (preference).
2. We need to place J, K, L in slots 1, 2, and 4.
3. We need at least two more preferences met. The preferences are L=1, K=2, J=1 or 2.
4. To get two preferences, we must use L=1 and K=2, OR L=1 and J=2.
- \textbfPossibility 1: L=1 (pref), K=2 (pref). This leaves J for slot 4 (not pref). Total preferences: L, K, N, M (4). The schedule is: L(1), K(2), N(3), J(4), M(5).
- \textbfPossibility 2: L=1 (pref), J=2 (pref). This leaves K for slot 4 (not pref, but acceptable). Total preferences: L, J, N, M (4). The schedule is: L(1), J(2), N(3), K(4), M(5).
5. Now we test the options against these two possible schedules:
- (A) J's appointment is 1. False in both possibilities.
- (B) N's appointment is 1. False, N is always 3.
- (C) J's appointment is earlier than K's. In Possibility 2, J is at 2 and K is at 4. So J is earlier than K. This can be true.
- (D) K's appointment is earlier than L's. False, L is always at 1 in these scenarios.
- (E) N's appointment is earlier than L's. False, L is at 1 and N is at 3.
Step 4: Final Answer:
The schedule L(1), J(2), N(3), K(4), M(5) is valid when M=5, and in this schedule, J's appointment is earlier than K's.

Answer 6

Step 1: Understanding the Concept:
We are given a relational condition (K>N) and asked for a necessary consequence.
Step 2: Key Formula or Approach:
Translate the condition into specific slots, then apply the preference rules to deduce the full arrangement.
Step 3: Detailed Explanation:
1. We know N is always in slot 3. The condition "K is later than N" means K must be in slot 4 or 5.
2. K's preference is slot 2. Since K is in 4 or 5, K does not get its preference.
3. Since exactly 4 patients must get their preference, and K is not one of them, the other four (J, L, N, M) must all get their preferences.
4. L prefers 1 \(\rightarrow\) L must be in slot 1.
5. J prefers 1 or 2. Since L is in 1, J must be in slot 2.
6. N prefers 3 \(\rightarrow\) N is in slot 3 (as always).
7. M prefers 4 or 5.
8. This gives us the partial schedule: L(1), J(2), N(3).
9. The remaining patients, K and M, must fill the remaining slots, 4 and 5. This is consistent with our deductions for K and M.
10. So, if K is later than N, the schedule must begin with L in slot 1 and J in slot 2.
11. Let's examine the options for what "must be true":
- (A) J is scheduled for appointment 4. False, J must be 2.
- (B) K is scheduled for appointment 5. False, K could be 4 and M could be 5. It's not a must.
- (C) L is scheduled for appointment 1. True, this was a necessary deduction.
- (D) M is scheduled for appointment 4. False, M could be 5 and K could be 4. It's not a must.
- (E) N is scheduled for appointment 2. False, N must be 3.
Step 4: Final Answer:
The condition that K is later than N forces a specific set of preferences to be met, which in turn fixes L's position to be in appointment 1.