To solve this problem, we need to calculate how many ways four couples can arrange themselves in a row, with the condition that each couple stays together but can be anywhere in the line.

Step 1: Treat each couple as a unit

Since the couples must stay together, we can treat each couple as a single "block" or unit. Therefore, instead of arranging 8 individuals (4 couples with 2 members each), we only need to arrange 4 units (the 4 couples).

The number of ways to arrange these 4 "blocks" (couples) in a row is simply the number of permutations of 4 items, which is given by:

$4! = 4 \times 3 \times 2 \times 1 = 24$

Step 2: Arrange the individuals within each couple

Each couple can be arranged internally in 2 ways (one person can be first, or the other person can be first). Since there are 4 couples, the number of ways to arrange the individuals within each couple is:

$2^4 = 2 \times 2 \times 2 \times 2 = 16$

Step 3: Total arrangements

To find the total number of ways to arrange the couples in a row, we multiply the number of ways to arrange the couples (4!) by the number of ways to arrange the individuals within each couple (2^4):

$4! \times 2^4 = 24 \times 16 = 384$

Thus, the total number of ways the four couples can arrange themselves in a row while staying together is 384.

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Would you like further details or clarifications on any step? Here are some related questions:

1. How would the answer change if there were 5 couples instead of 4?
2. How do we calculate the number of ways to arrange people in a row if no restrictions are applied?
3. How does the concept of "permutations" apply to this problem?
4. What happens if each couple has to sit next to their partner, but not necessarily together?
5. How can the answer be calculated if the couples were not restricted to staying together?

Tip: When dealing with arrangements involving multiple groups, it’s often helpful to treat each group as a unit first and then calculate the internal arrangements later.