Five students, Adnan, Beth, Chao, Dan, and Edmund are to be arranged in a line. How many such arrangements are possible if Beth is not allowed to stand next to dan? 

To solve this problem, we need to calculate the total number of ways to arrange five students (Adnan, Beth, Chao, Dan, and Edmund) with the restriction that Beth and Dan must not stand next to each other.

### Step 1: Calculate the total number of arrangements without any restrictions.
In general, for 5 distinct students, the number of possible arrangements is the number of permutations of 5 items, which is:
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 2: Calculate the number of arrangements where Beth and Dan are together.
To find how many ways Beth and Dan can stand next to each other, treat them as a single "block" or unit. Now, instead of 5 individuals, we have 4 units to arrange: (Beth-Dan), Adnan, Chao, and Edmund.

The number of ways to arrange 4 units is:
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
Within this "block" or unit, Beth and Dan can switch places, so for each arrangement of 4 units, Beth and Dan can be arranged in \(2!\) ways (since Beth and Dan can switch spots within their block):
\[
2! = 2 \times 1 = 2
\]
Therefore, the total number of ways Beth and Dan can be next to each other is:
\[
4! \times 2! = 24 \times 2 = 48
\]

### Step 3: Subtract the restricted cases from the total cases.
Now, subtract the number of arrangements where Beth and Dan are together from the total number of arrangements:
\[
120 - 48 = 72
\]

Thus, there are **72** possible arrangements where Beth and Dan are not standing next to each other.

### Would you like further details or have any questions?

#### Related questions:
1. What is the probability that Beth and Dan will stand next to each other in a random arrangement?
2. How would the solution change if another pair of students must not stand together as well?
3. If we add another student, how does the solution adjust?
4. How many arrangements are possible if Adnan must always stand at the beginning?
5. How many ways can Beth and Dan be arranged if they must be at opposite ends?

#### Tip: When dealing with permutation problems, breaking the problem into smaller cases (like grouping units) is often helpful.

Five students, Adnan, Beth, Chao, Dan, and Edmund are to be arranged in a line. How many such arrangements are possible if Beth is not allowed to stand next to Dan?

This solution explains how to calculate the number of possible arrangements for five students with the restriction that Beth cannot stand next to Dan. It uses combinatorics and permutations principles, making it suitable for students in Grades 9-12. The solution includes step-by-step calculations using factorials and restricted arrangement formulas.

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