A family consisting of two parents and four
children is to be seated in a row for a picture. How many dif-ferent arrangements are possible in which the children are
seated together?

To solve the problem, we'll break it down into steps:

### Step 1: Treat the four children as a single unit
Since the four children must be seated together, we can treat them as a single block or unit. This changes the problem from arranging 6 people (2 parents and 4 children) to arranging 3 units (2 parents and 1 block of children).

### Step 2: Arrange the units
There are \(3!\) (3 factorial) ways to arrange the 3 units (2 parents and 1 block of children).

\[
3! = 3 \times 2 \times 1 = 6
\]

### Step 3: Arrange the children within their block
Within the block of children, the 4 children can be arranged in \(4!\) (4 factorial) different ways.

\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

### Step 4: Multiply the results
To find the total number of different arrangements, we multiply the number of ways to arrange the units by the number of ways to arrange the children within their block.

\[
\text{Total arrangements} = 3! \times 4! = 6 \times 24 = 144
\]

### Final Answer:
There are **144** different arrangements in which the children are seated together.

Would you like any more details or have any questions?

Here are some related questions to consider:
1. How many different arrangements are possible if no restrictions are placed on seating?
2. What if the parents must also sit together, how many arrangements are possible?
3. How many different arrangements are possible if exactly two of the children must sit together?
4. If the four children were not all to be seated together, how would you calculate the number of arrangements?
5. How would the answer change if there were three parents and four children?

**Tip:** When solving combinatorial problems, it's helpful to consider treating groups of items as single units to simplify the problem.

Discover how many different arrangements are possible for seating a family of two parents and four children in a row for a picture. Learn the step-by-step solution using combinatorial methods, including permutations and factorials.

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