Question

Three husband–wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?

• A. 16
• B. 4
• C. 120
• D. 720

Hint:

For "couples together" at a round table: first arrange couple-blocks in $(k-1)!$ ways, then multiply by $2^k$ for within-couple orderings (assuming only rotations are identical).

Answer 1

The Correct Option is A

Step 1: Treat each couple as a block.
If each husband must sit next to his wife, consider each couple as a single block. There are $3$ blocks to arrange around a circle. For circular arrangements (only rotations considered identical), the number of ways to arrange $3$ distinct blocks is $(3-1)! = 2$.

Step 2: Arrange within each block.
Inside a couple's block, the two people can sit as $(H,W)$ or $(W,H)$, i.e., $2$ ways per block.
Hence internal arrangements contribute a factor of $2^3 = 8$.

Step 3: Multiply choices.
Total valid seatings $= (3-1)! \times 2^3 = 2 \times 8 = 16$.
\[ \boxed{16} \]