Puzzle 3: The Council’s Circular Table

The Archivist Council consists of 8 members, who must sit around a circular table. There are strict seating rules:
Alice and Bob must NOT sit next to each other.
Ali and Ranjid must sit next to each other.
Since this is a circular table, there is no fixed starting point for the seating arrangement.
Question:
How many distinct ways can the council members be seated around the table under these conditions?

To solve this problem, we need to calculate the number of distinct seating arrangements for the 8 members of the council under the given conditions:

### Step 1: General Principle for Circular Permutations
For circular permutations of \( n \) people, the number of distinct ways to arrange them around a table is \((n - 1)!\). This is because in a circular arrangement, one person's position can be fixed to remove rotational symmetry, and the remaining \( n - 1 \) people can be arranged in a linear sequence.

In this case, we have 8 council members, so without any seating restrictions, the total number of ways to arrange them would be:
\[
(8 - 1)! = 7! = 5040
\]
However, there are seating constraints that we must account for.

### Step 2: Handling the Condition that Ali and Ranjid Must Sit Together
We can treat Ali and Ranjid as a single "block" or "unit" since they must always sit next to each other. Now, instead of 8 separate members, we have 7 units (6 individual members plus the Ali-Ranjid block).

The number of ways to arrange these 7 units around a circular table is \((7 - 1)! = 6! = 720\).

Within the block, Ali and Ranjid can switch places with each other, so there are 2 possible arrangements for the block. Therefore, the total number of ways to arrange the council members, accounting for Ali and Ranjid sitting together, is:
\[
6! \times 2 = 720 \times 2 = 1440
\]

### Step 3: Handling the Condition that Alice and Bob Must Not Sit Next to Each Other
Now, we need to subtract the cases where Alice and Bob are seated next to each other. To do this, we will calculate how many of the 1440 arrangements have Alice and Bob sitting next to each other and subtract those from the total.

#### Step 3.1: Treating Alice and Bob as a Block
If Alice and Bob are also required to sit next to each other, we can treat them as a block. Now we have 6 units: the Ali-Ranjid block, the Alice-Bob block, and 4 other members.

The number of ways to arrange these 6 units around the circular table is \((6 - 1)! = 5! = 120\).

Within their block, Alice and Bob can switch places, so there are 2 possible arrangements for the Alice-Bob block. Also, within the Ali-Ranjid block, Ali and Ranjid can switch places, so there are 2 arrangements for their block as well.

Thus, the total number of ways to arrange the council members with both Alice and Bob sitting next to each other and Ali and Ranjid sitting next to each other is:
\[
5! \times 2 \times 2 = 120 \times 2 \times 2 = 480
\]

#### Step 3.2: Subtracting the Unwanted Cases
To find the number of valid seating arrangements where Alice and Bob are **not** sitting next to each other, we subtract the cases where they are sitting next to each other from the total number of arrangements:
\[
1440 - 480 = 960
\]

### Step 4: Conclusion
The total number of distinct ways to seat the council members around the table, such that Ali and Ranjid are next to each other and Alice and Bob are not next to each other, is:
\[
\boxed{960}
\]

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Do you want details on any specific step or have questions?

Here are some related problems to expand your understanding:

1. How would the solution change if the council had 9 members instead of 8?
2. What if Ali and Ranjid must sit *opposite* each other instead of next to each other?
3. How many ways can Alice and Bob be seated *exactly* next to each other, without other restrictions?
4. What if Alice and Bob must be separated by exactly two people at the table?
5. How would you solve this if the table were not circular but linear?

**Tip:** In combinatorics, treating people or objects that must stay together as a "block" is a powerful strategy to simplify complex seating or arrangement problems.

Puzzle 3: The Council’s Circular Table. The Archivist Council consists of 8 members, who must sit around a circular table. There are strict seating rules: Alice and Bob must NOT sit next to each other. Ali and Ranjid must sit next to each other. Since this is a circular table, there is no fixed starting point for the seating arrangement. How many distinct ways can the council members be seated around the table under these conditions?

Explore how to solve a combinatorics problem involving circular permutations. Learn how to arrange 8 members of a council around a circular table with constraints: Alice and Bob must not sit next to each other, and Ali and Ranjid must sit together. This detailed solution covers circular permutation formulas and the inclusion-exclusion principle, suitable for Grades 10-12.

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