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This image contains a problem written in Japanese about arranging five people (A, B, C, D, and E) in a row. Here is the translation and explanation of the three questions in the image:

1. **(1) How many ways can the arrangement be made if A is at one end?**
2. **(2) How many ways can the arrangement be made if B and C are next to each other?**
3. **(3) How many ways can the arrangement be made if B and C are not next to each other?**

### Solutions:

#### (1) A is at one end:
If A must be at one end (either the first or the last position), then we fix A in one of the two end positions. The remaining four people (B, C, D, E) can be arranged in the remaining 4 positions. The number of ways to arrange 4 people is \(4! = 24\). Since A can be at either end (two options), the total number of arrangements is:
\[
2 \times 4! = 2 \times 24 = 48 \text{ ways}.
\]

#### (2) B and C are next to each other:
We can treat B and C as a single block (BC or CB), so we are now arranging 4 blocks: (BC), A, D, E. The number of ways to arrange these 4 blocks is \(4! = 24\). Since B and C can switch places within the block (BC or CB), we have 2 ways to arrange them within the block. Therefore, the total number of ways is:
\[
4! \times 2 = 24 \times 2 = 48 \text{ ways}.
\]

#### (3) B and C are not next to each other:
The total number of arrangements without restrictions is \(5! = 120\). From this, we subtract the number of ways in which B and C are next to each other (calculated in part (2), which is 48). Therefore, the number of ways in which B and C are not next to each other is:
\[
5! - 48 = 120 - 48 = 72 \text{ ways}.
\]

### Final Answers:
1. \( 48 \text{ ways} \)
2. \( 48 \text{ ways} \)
3. \( 72 \text{ ways} \)

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### Related Questions:
1. How many ways can the five people be arranged without any restrictions?
2. If A must be in the center, how many different arrangements are possible?
3. How many ways can the arrangement be made if D and E must be next to each other?
4. If A and B cannot be next to each other, how many different arrangements are possible?
5. How many ways can the arrangement be made if B and C must be at the ends?

### Tip:
When solving arrangement problems involving "next to each other" conditions, try grouping the elements into a single block for simplicity.

A, B, C, D, and E are arranged in a row. Answer the following questions:

1. How many ways can they be arranged if A is at one end?
2. How many ways can they be arranged if B and C are next to each other?
3. How many ways can they be arranged if B and C are not next to each other?

Explore combinatorics with this problem involving the arrangement of five individuals (A, B, C, D, E) in a row under different conditions. Learn how to calculate arrangements when specific conditions, such as one person at the end or two people next to each other, are applied. Suitable for high school students studying permutations and combinations.

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