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.