GATE Civil 2023 Set 2 | GA | Question: 7

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. $720$

Comments

For those who doesn’t know: It is a question from the concept of circular arrangements.

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some variations : 

1. Five husband-wife pairs are to be seated at a round table that has ten 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? 

2. Two husband-wife pairs and one individual are to be arranged at a circular table with six identical chairs. If the seating arrangements consider only the relative positions, how many arrangements can be made with the condition that each husband sits next to his wife?

Solution

Answer 1

Every husband sits next to his wife, so we can say every husband-wife is a one-block.

There are $3$ husband-wife pairs, and each pair can sit in $2$ ways (Husband-Wife or Wife-Husband). The total number of seating arrangements is $2^3 = 8$ for each pair.

Now, consider these $3$ pairs as one unit each and arrange them in a circle, which can be done in $(3-1)!$ ways.

So, the total seating arrangements $= (3-1)! * 2^3 = 2 * 8 = 16.$

Correct Answer: A

Comments

@Lakshman Bhaiya Why did we do (3-1)! ?

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@Sachi Mathur

No. of ways to arrange n objects in circular way = (n-1)!

Answer 2

ANS A.

1. Treat each couple as a single unit: Since every husband must sit next to his wife, treat each husband-wife pair as a single "block." So, we now have 3 blocks to arrange.

2. Arrange the 3 blocks: In a circular arrangement, fixing one block in one position eliminates rotation. This leaves 2 blocks, which can be arranged in 2!=2 ways.

3. Arrange within each block: Within each block (couple), the husband and wife can switch seats. So, for each of the 3 blocks, there are 2 ways to arrange the husband and wife, which gives 2^3=8 ways.

4. Total arrangements: Multiply the ways to arrange the blocks and the ways to arrange within blocks:

2!×2^3=2×8=16.

So, there are 16 possible seating arrangements.