An automobile manufacturer offers a station wagon with either a 6-cylinder engine or a 4-cylinder engine and with either a...

1. Translate the problem requirements: Identify all available options and any restrictions that limit certain combinations
2. Identify the constraint structure: Recognize that the trailer hitch restriction creates two separate scenarios to analyze
3. Enumerate valid combinations systematically: List all possible combinations for each engine type, respecting the trailer hitch constraint
4. Count total combinations: Sum up all valid combinations from both scenarios

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by clearly understanding what options are available and what restrictions exist.

The manufacturer offers these basic options:
• Engine type: 6-cylinder OR 4-cylinder
• Transmission type: Manual OR Automatic
• Trailer hitch: Available OR Not available

Now here's the key restriction we need to understand: The trailer hitch is ONLY offered on station wagons with a 6-cylinder engine. This means if you choose a 4-cylinder engine, you cannot get a trailer hitch at all.

Process Skill: TRANSLATE - Converting the problem's business language into clear mathematical constraints

2. Identify the constraint structure

The trailer hitch restriction creates two completely separate scenarios that we need to analyze:

Scenario A: Station wagons with 6-cylinder engines
• Can have manual OR automatic transmission
• Can have trailer hitch OR no trailer hitch

Scenario B: Station wagons with 4-cylinder engines
• Can have manual OR automatic transmission
• Cannot have trailer hitch (restriction applies)

This constraint structure tells us we cannot simply multiply all options together - we must consider each engine type separately.

Process Skill: APPLY CONSTRAINTS - Recognizing how restrictions limit our combinations

3. Enumerate valid combinations systematically

Now let's list every valid combination for each scenario:

Scenario A (6-cylinder engine):

1. 6-cylinder + Manual + No trailer hitch
2. 6-cylinder + Manual + Trailer hitch
3. 6-cylinder + Automatic + No trailer hitch
4. 6-cylinder + Automatic + Trailer hitch

Scenario B (4-cylinder engine):

5. 4-cylinder + Manual + No trailer hitch
6. 4-cylinder + Automatic + No trailer hitch

Notice that we cannot have "4-cylinder + trailer hitch" combinations because of the restriction.

Process Skill: CONSIDER ALL CASES - Systematically examining each valid scenario

4. Count total combinations

Counting our enumerated combinations:
• Scenario A (6-cylinder): 4 valid combinations
• Scenario B (4-cylinder): 2 valid combinations
• Total: 4 + 2 = 6 combinations

Verifying against our answer choices: 6 corresponds to choice D.

Final Answer

The manufacturer offers 6 different combinations for its station wagon. The answer is D.

Common Faltering Points

Errors while devising the approach

1. Misreading the constraint scope
Students often misread "trailer hitch is only offered on station wagons with 6-cylinder engines" as "trailer hitch is required on all 6-cylinder engines." This leads them to think that every 6-cylinder wagon must have a trailer hitch, when actually it's optional for 6-cylinder engines and forbidden for 4-cylinder engines.

2. Attempting to use simple multiplication without considering constraints
Many students see "\(2 \times 2 \times 2 = 8\) combinations" and select answer E without recognizing that the trailer hitch constraint makes this multiplication approach invalid. They fail to realize that not all theoretical combinations are actually available.

3. Misunderstanding what constitutes a "combination"
Some students get confused about whether "no trailer hitch" counts as an option or if they should only count configurations that include a trailer hitch. They may think there are only 4 combinations (the ones with hitches) rather than recognizing that both "with hitch" and "without hitch" are valid choices for 6-cylinder engines.

Errors while executing the approach

1. Forgetting to include "no trailer hitch" combinations
Even when students correctly identify the constraint, they sometimes only enumerate the combinations that include a trailer hitch (2 combinations for 6-cylinder engines) and forget to count the combinations where customers choose no trailer hitch, leading to an undercount.

2. Double-counting or missing combinations during enumeration
When listing combinations systematically, students may accidentally count the same combination twice (especially when switching between engine types) or skip valid combinations, particularly the "no hitch" options for 6-cylinder engines.

Errors while selecting the answer

No likely faltering points