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... 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?
Understand the Problem
The question is asking how many seating arrangements can be made for three husband-wife pairs at a circular table, under the condition that each husband must sit next to his wife. This involves a combinatorial problem related to arrangements in a circular format.
Answer
The total number of seating arrangements is \( 16 \).
Answer for screen readers
The total number of seating arrangements is ( 16 ).
Steps to Solve
- Group the husband-wife pairs
Since each husband must sit next to his wife, we can treat each pair as a single unit. For three pairs, we have three "blocks" to arrange: (H1, W1), (H2, W2), and (H3, W3).
- Calculate arrangements of the blocks
In a circular arrangement, the number of ways to arrange $n$ items is given by $(n-1)!$. For our three pairs, we can arrange them in:
$$(3-1)! = 2! = 2 \text{ arrangements of pairs}$$
- Calculate arrangements within each pair
Each pair can be arranged in 2 ways (husband can sit on the left or right of the wife). For three pairs, this amounts to:
$$2^3 = 8 \text{ arrangements within pairs}$$
- Combine arrangements
Now we multiply the number of arrangements of blocks by the arrangements within each pair:
$$\text{Total arrangements} = (2 \text{ arrangements of pairs}) \times (8 \text{ arrangements within pairs}) = 2 \times 8 = 16$$
The total number of seating arrangements is ( 16 ).
More Information
This problem illustrates the concept of permutations in combinatorial settings, especially in circular arrangements. By treating husband-wife pairs as single units, we simplify the arrangement process significantly.
Tips
- Ignoring the circular arrangement: Always remember that in a circular seating, one position is fixed to avoid counting the same arrangement multiple times.
- Not considering the internal arrangement of pairs: Each pair can be arranged in two ways, which is crucial to the final count.
AI-generated content may contain errors. Please verify critical information
