A group of 10 friends consists of 5 married couples. How many seating arrangements are possible for these 10 friends...

1. Translate the problem requirements: We need to count seating arrangements where 10 people (5 married couples) sit in a row, with the constraint that each person must sit next to their spouse
2. Recognize the constraint structure: Since each person must sit next to their spouse, we can treat each married couple as a single unit that must stay together
3. Count arrangements of couple-units: Determine how many ways we can arrange the 5 couple-units in the row
4. Account for internal arrangements: For each couple-unit, count how many ways the two spouses can be arranged within their unit
5. Apply multiplication principle: Multiply the number of ways to arrange units by the number of internal arrangements per unit

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're asked to find. We have 10 people total - these people form 5 married couples. We want to seat all 10 people in a row of 10 seats, where each seat holds exactly one person.

The key constraint is: each person must be seated next to his or her spouse. This means that in our final seating arrangement, every married person must have their spouse in the seat immediately to their left or right.

Process Skill: TRANSLATE - Converting the constraint "next to spouse" into the mathematical requirement that couples must be adjacent

2. Recognize the constraint structure

Here's the key insight: since each person must sit next to their spouse, we can think of each married couple as a single "block" or "unit" that cannot be separated.

Instead of thinking about arranging 10 individual people, we can think about arranging 5 couple-blocks. Each couple-block takes up exactly 2 seats, and the 5 blocks will fill all 10 seats.

This transforms our problem from "arrange 10 people with constraints" to "arrange 5 units without constraints."

3. Count arrangements of couple-units

Now we ask: in how many ways can we arrange 5 couple-units in a row?

This is straightforward - we have 5 distinct units (Couple A, Couple B, Couple C, Couple D, Couple E), and we want to arrange them in 5 positions.

The first position can be filled by any of the 5 couples.
The second position can be filled by any of the remaining 4 couples.
The third position can be filled by any of the remaining 3 couples.
The fourth position can be filled by any of the remaining 2 couples.
The fifth position must be filled by the last remaining couple.

So we have: \(5 \times 4 \times 3 \times 2 \times 1 = 5! = 120\) ways to arrange the couple-units.

4. Account for internal arrangements

But wait - we're not done yet! Within each couple-unit, the two spouses can be arranged in different ways.

For any given couple, let's call them Husband and Wife. They can sit as:
- Husband on the left, Wife on the right, OR
- Wife on the left, Husband on the right

So each couple-unit has exactly 2 internal arrangements.

Since we have 5 couples, and each couple independently has 2 internal arrangements:
Total internal arrangements = \(2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32\)

5. Apply multiplication principle

Now we combine our results using the multiplication principle:

Total seating arrangements = (Ways to arrange couple-units) × (Internal arrangements)

Total seating arrangements = \(120 \times 32 = 3,840\)

Process Skill: APPLY CONSTRAINTS - Systematically accounting for both the positioning of couples and the internal arrangement within each couple

Final Answer

The total number of seating arrangements is 3,840.

Looking at our answer choices:

1. 45
2. 120
3. 252
4. 3,840
5. 14,400

Our answer matches choice D. 3,840.

Common Faltering Points

Errors while devising the approach

1. Failing to recognize the "blocking" concept: Many students attempt to arrange all 10 individuals separately while trying to manually ensure couples stay together. This leads to an extremely complex counting process with overlapping cases. The key insight is recognizing that "each person must sit next to their spouse" means couples form inseparable blocks that should be treated as single units.

2. Misinterpreting the constraint: Students may misread "next to his or her spouse" as meaning couples can sit anywhere as long as they're in the same row, rather than understanding it requires immediate adjacency (no other person between spouses). This fundamental misunderstanding leads to counting many invalid arrangements.

3. Overlooking the two-step nature: Students often recognize the blocking concept but fail to realize there are TWO separate counting steps: (1) arranging the 5 couple-blocks, and (2) arranging people within each block. They may only count one aspect, typically just the \(5! = 120\) arrangements of couples, missing the internal arrangements entirely.

Errors while executing the approach

1. Incorrect calculation of internal arrangements: Students may incorrectly calculate the number of ways to arrange people within couples. Common errors include thinking there's only 1 way (forgetting that husband-wife vs wife-husband are different), or miscounting the total combinations across all couples (perhaps calculating 2×5 = 10 instead of \(2^5 = 32\)).

2. Arithmetic errors in final multiplication: When combining the two parts, students may make computational mistakes in calculating \(120 \times 32\). Common errors include: \(120 \times 32 = 3,640\) (missing a digit) or \(120 \times 30 = 3,600\) (rounding 32 to 30), leading them to select incorrect answer choices.

Errors while selecting the answer

1. Selecting partial results: Students who correctly identify the two-step process but make execution errors often end up with intermediate values that match wrong answer choices. For example, if they only count couple arrangements (120) or make arithmetic mistakes, they might select choice B (120) thinking they've solved the complete problem, when they've only completed the first step.