How many ways are there for   c and f to sit next to each other and a and b to be opposite

To solve this problem, we need to determine the number of ways \( c \) and \( f \) can sit next to each other, and \( a \) and \( b \) can sit opposite each other. Let’s break it down step by step:

### Assumptions:
- There are 6 people: \( a, b, c, d, e, f \), sitting around a **circular table**.
- We treat two seats as "opposite" if they are directly across from each other.

### Step-by-step solution:

#### Step 1: Fix \( a \) and \( b \) opposite each other
Since \( a \) and \( b \) must be opposite each other, we first select two seats for them. In a circular arrangement, fixing the position of one person (say, \( a \)) makes the arrangement linear, and the seat directly opposite is fixed for \( b \). So there’s only **1 way** to arrange \( a \) and \( b \) opposite each other.

#### Step 2: Treat \( c \) and \( f \) as a block
Since \( c \) and \( f \) must sit next to each other, treat them as a single "block" or "unit." Now, instead of arranging 4 individual people, we are arranging 5 units (the \( c \)-\( f \) block, and the remaining 3 people \( d, e \), and one additional person).

For these 5 units, we can arrange them in **\( 3! \)** (factorial) ways, since we have 3 remaining people plus the \( c \)-\( f \) block.

#### Step 3: Arrange \( c \) and \( f \) within their block
Within the block, \( c \) and \( f \) can sit in **2 ways**: either \( (c, f) \) or \( (f, c) \).

### Total number of arrangements:
- 1 way to arrange \( a \) and \( b \) opposite each other.
- \( 3! \) ways to arrange the 3 remaining people and the \( c \)-\( f \) block: \( 3! = 6 \).
- 2 ways to arrange \( c \) and \( f \) within their block.

Thus, the total number of ways is:
\[
1 \times 6 \times 2 = 12 \text{ ways}.
\]

### Conclusion:
There are **12 ways** for \( c \) and \( f \) to sit next to each other and \( a \) and \( b \) to sit opposite each other.

Do you want further details or explanations?

Here are some related questions:
1. How would the answer change if it wasn't a circular table?
2. How many ways are there if \( d \) and \( e \) must sit next to each other instead?
3. What if \( a \), \( b \), and \( c \) need to be in specific seats, but not necessarily opposite?
4. How many seating arrangements are there if only \( a \) and \( b \) must sit together?
5. What if \( c \) and \( f \) cannot sit next to each other?

**Tip:** In circular arrangements, fixing one person's position helps avoid counting equivalent rotations multiple times.

How many ways are there for c and f to sit next to each other and a and b to be opposite?

Explore how to calculate the number of ways for two people to sit next to each other and two others to sit opposite each other in a circular seating arrangement. This solution covers concepts like permutations, circular arrangements, and factorial calculations, suitable for Grades 9-12.

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