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?

Question

Prepp Team · Accepted Answer

Circular Arrangements with Pair Constraint

The problem asks for the number of ways to seat three husband-wife pairs around a circular table such that each husband sits next to his wife. Since arrangements are based on relative positions, we use circular permutation principles.

Step 1: Treat Pairs as Units

The core constraint is that each husband must sit next to his wife. This means we can consider each husband-wife pair as a single, indivisible unit.

Let the pairs be P1, P2, and P3.
We now need to arrange these 3 distinct units around a circular table.

Step 2: Circular Arrangement of Units

The number of ways to arrange $n$ distinct items in a circle is given by the formula $(n-1)!$.

In this case, $n=3$ (the three pairs).
Number of ways to arrange the 3 pairs = $(3-1)! = 2! = 2$.

Step 3: Internal Arrangement within Pairs

Within each pair unit (e.g., Husband A and Wife A), the two individuals can switch positions.

For Pair 1, there are 2 possible arrangements: (Husband1, Wife1) or (Wife1, Husband1).
Similarly, for Pair 2, there are 2 arrangements.
And for Pair 3, there are 2 arrangements.
Total internal arrangements for all 3 pairs = $2 \times 2 \times 2 = 2^3 = 8$.

Step 4: Total Seating Arrangements

To find the total number of possible seating arrangements satisfying the condition, we multiply the number of ways to arrange the units around the table by the number of ways to arrange individuals within each unit.