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.