A group of five people consisting of a couple are to be seated on a round. table for a meeting. What is the total. number of ways in which the seating arrangement can be made so that the couple do NOT sit next to each other?

The Correct answer is Option 3. 

Key Points

Total Arrangements Without Any Restriction:

⇒ In a circular arrangement, the number of ways to arrange  n people is (n−1)!.

⇒ For 5 people, the number of circular arrangements is  (5−1)! = 4! = 24 ways.

Arrangements Where the Couple Sits Together:

⇒ Consider the couple as a single unit. Now we have 4 units to arrange (the couple and the other 3 individuals).
⇒ The number of ways to arrange these 4 units in a circle is  (4−1)!=3!=6 ways.
⇒ Within the couple's unit, the two people can switch places, so there are 2 additional arrangements.
⇒ Therefore, the number of ways in which the couple sits together is  6×2=12.
Arrangements Where the Couple Does Not Sit Together:

⇒ To find this, subtract the number of arrangements where the couple sits together from the total number of arrangements. 24−12=12.
⇒ Thus, the total number of ways to arrange the group such that the couple does not sit next to each other is 12. Hence, Option 3 is correct.