Question Analysis

The question asks us to find a possible order in which a medical representative can visit six hospitals — A, B, C, D, E, and F — exactly once in a day, while adhering to specific scheduling constraints.

Understanding the Constraints

There are three main conditions that must be satisfied for the medical representative's schedule:

• Constraint i: He must visit A before B and E. This means that in the sequence of visits, A must appear earlier than both B and E. We can represent this as $A \rightarrow B$ and $A \rightarrow E$.
• Constraint ii: He must visit B before D. In the sequence, B must appear earlier than D. We can represent this as $B \rightarrow D$.
• Constraint iii: The third hospital he visits must be C. This fixes the position of C in the schedule. The order must look like $\text{_ _ C _ _ _}$.

Combining Constraints

From Constraint i ($A \rightarrow B$) and Constraint ii ($B \rightarrow D$), we can infer that A must come before B, and B must come before D. This establishes a relative order of $A \rightarrow B \rightarrow D$. Additionally, Constraint i tells us $A \rightarrow E$. Therefore, the overall relative order constraints are $A \rightarrow B$, $A \rightarrow E$, and $B \rightarrow D$. C must always be in the third position.

Evaluating Each Possible Order

We need to check each of the given options to see if it satisfies all three constraints simultaneously.

Option 1: A, F, C, D, E, B

Let's check the constraints for this order:

• Constraint i ($A \rightarrow B$, $A \rightarrow E$): A is visited first, B is visited last, and E is visited fifth. A is before B and A is before E. This constraint is satisfied.
• Constraint ii ($B \rightarrow D$): B is visited last (6th) and D is visited fourth (4th). B is visited after D, not before. This constraint is not satisfied.
• Constraint iii (Third is C): The third hospital visited is indeed C. This constraint is satisfied.

Since Constraint ii is not satisfied, this option is not a possible order.

Option 2: C, E, A, B, D, F

Let's check the constraints for this order:

• Constraint i ($A \rightarrow B$, $A \rightarrow E$): A is visited third, B is visited fourth, and E is visited second. A is before B (3rd before 4th), but A is not before E (3rd is not before 2nd). This constraint is not satisfied.
• Constraint ii ($B \rightarrow D$): B is visited fourth, and D is visited fifth. B is before D. This constraint is satisfied.
• Constraint iii (Third is C): The third hospital visited is A, not C. This constraint is not satisfied.

Since Constraints i and iii are not satisfied, this option is not a possible order.

Option 3: C, F, A, E, D, B

Let's check the constraints for this order:

• Constraint i ($A \rightarrow B$, $A \rightarrow E$): A is visited third, B is visited sixth, and E is visited fourth. A is before B (3rd before 6th), and A is before E (3rd before 4th). This constraint is satisfied.
• Constraint ii ($B \rightarrow D$): B is visited sixth, and D is visited fifth. B is visited after D, not before. This constraint is not satisfied.
• Constraint iii (Third is C): The third hospital visited is A, not C. This constraint is not satisfied.

Since Constraints ii and iii are not satisfied, this option is not a possible order.

Option 4: A, B, C, D, E, F

Let's check the constraints for this order:

• Constraint i ($A \rightarrow B$, $A \rightarrow E$): A is visited first, B is visited second, and E is visited fifth. A is before B (1st before 2nd), and A is before E (1st before 5th). This constraint is satisfied.
• Constraint ii ($B \rightarrow D$): B is visited second, and D is visited fourth. B is before D (2nd before 4th). This constraint is satisfied.
• Constraint iii (Third is C): The third hospital visited is indeed C. This constraint is satisfied.

Since all three constraints are satisfied, this option is a possible order.

Identifying the Correct Order

Based on the evaluation of each option against the given constraints, only the order A, B, C, D, E, F satisfies all the conditions for the medical representative's hospital visits.

Revision Table: Checking Constraints for Hospital Visit Orders

Additional Information on Scheduling Problems and Constraints

Scheduling problems involve arranging a set of tasks or events in a specific order, often subject to various restrictions or rules called constraints. These problems are common in logistics, project management, and daily planning.

Key concepts related to this type of problem include:

• Precedence Constraints: These specify that one task or event must be completed before another can begin. In this problem, $A \rightarrow B$, $A \rightarrow E$, and $B \rightarrow D$ are precedence constraints. They dictate the relative order of certain hospitals.
• Fixed Position Constraints: These require a specific task or event to occur at a particular point in the schedule. The condition that the third hospital visited must be C is a fixed position constraint.
• Permutations: A permutation is an arrangement of all members of a set into a sequence or a linear order. In this problem, finding a possible schedule means finding a valid permutation of the six hospitals that satisfies all the given constraints. The total number of possible permutations of 6 hospitals is $6! = 720$, but the constraints significantly reduce the number of valid possibilities.

Solving scheduling puzzles like this involves systematically applying the constraints to eliminate invalid arrangements until only those that meet all conditions remain. Testing the given options against each constraint is an effective method for such multiple-choice questions.