Understanding the Team Selection Problem

We are given a group of seven men: A, B, C, D, E, F, and G, from which we need to select a team of exactly four men. The selection must adhere to specific rules or constraints.

Specific Constraints for Team Formation

The following conditions must be met when forming the team of four:

• Constraint 1: Men A and B must always be selected together in the team.
• Constraint 2: Men D and F must always be selected together in the team.
• Constraint 3: Men C and E cannot be selected together in the same team. If one is selected, the other must not be.
• Constraint 4: Men F and G cannot be selected together in the same team. If one is selected, the other must not be.

Our task is to identify which of the given options represents a possible team of four men that satisfies all these constraints simultaneously.

Analyzing Each Team Option Against Constraints

Let's check each provided option to see if it violates any of the given constraints.

Option 1: A, B, C and D

• Check Constraint 1 (A & B together): Yes, A and B are both in the team.
• Check Constraint 2 (D & F together): No, D is in the team, but F is not. Constraint 2 is violated.

Since Constraint 2 is violated, this team is not a possible selection.

Option 2: A, B, D and F

• Check Constraint 1 (A & B together): Yes, A and B are both in the team.
• Check Constraint 2 (D & F together): Yes, D and F are both in the team.
• Check Constraint 3 (C & E not together): Neither C nor E are in the team. Constraint 3 is not violated.
• Check Constraint 4 (F & G not together): F is in the team, but G is not. Constraint 4 is not violated.

All four constraints are satisfied by this team composition. Therefore, this team is possible.

Option 3: A, B, C and E

• Check Constraint 1 (A & B together): Yes, A and B are both in the team.
• Check Constraint 2 (D & F together): Neither D nor F are in the team. Constraint 2 is not violated.
• Check Constraint 3 (C & E not together): No, both C and E are in the team. Constraint 3 is violated.

Since Constraint 3 is violated, this team is not a possible selection.

Option 4: A, B, D and G

• Check Constraint 1 (A & B together): Yes, A and B are both in the team.
• Check Constraint 2 (D & F together): No, D is in the team, but F is not. Constraint 2 is violated.

Since Constraint 2 is violated, this team is not a possible selection. Note that including D implies F must be present (Constraint 2), and having F and G together is forbidden (Constraint 4), so having D and G together would indirectly violate Constraint 4 as well, but the direct violation of Constraint 2 is sufficient to rule out this option.

Conclusion: Identifying the Possible Team

After evaluating each option against all the given constraints, only the team composed of A, B, D, and F meets all the required conditions for selection.

Revision Table: Checking Constraints for Each Option

Additional Information: Combination Problems with Rules

Problems like this involve selecting a subset of items (the team members) from a larger set, subject to specific rules. These rules can dictate who must be together, who cannot be together, or other conditions.

• Solving such problems requires careful listing of all constraints.
• Then, each potential combination (in this case, the options provided) must be systematically checked against every single constraint.
• A combination is valid only if it satisfies every constraint.
• If a constraint involves a pair that must be together (like A and B), they act as a single unit in terms of mandatory inclusion, though they occupy two spots in the team.
• If a constraint involves a pair that cannot be together (like C and E), the presence of one person in the team automatically forbids the presence of the other.

This systematic process of checking is crucial for accurately solving selection problems with multiple conditions.