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 ?
[amp_mcq option1=”24″ option2=”18″ option3=”12″ option4=”6″ correct=”option3″]
Total number of ways to arrange 5 distinct people around a round table is (5-1)! = 4! = 24.
We want to find the number of ways the couple do NOT sit next to each other. This can be found by subtracting the number of arrangements where the couple *do* sit together from the total number of arrangements.
To find the number of arrangements where the couple sit together, treat the couple as a single unit.
Now we are arranging 4 units around the table: the couple unit + the other 3 individuals.
The number of ways to arrange these 4 units around a round table is (4-1)! = 3! = 6.
Within the couple unit, the two individuals (let’s call them P1 and P2) can sit in two ways: P1-P2 or P2-P1. This can be done in 2! = 2 ways.
So, the total number of arrangements where the couple sit together is the number of ways to arrange the 4 units multiplied by the number of ways the couple can be arranged within their unit: 6 * 2 = 12.
The number of ways the couple do NOT sit next to each other is:
Total arrangements – Arrangements where the couple sit together
= 24 – 12 = 12.
For round table arrangements, the total ways for n people is (n-1)!. The ways for a specific couple to sit together is (n-2)! * 2!. The ways for them not to sit together is (n-1)! – (n-2)! * 2! = (n-1)(n-2)! – 2(n-2)! = (n-1-2)(n-2)! = (n-3)(n-2)!.
In this case, n=5.
Total round table arrangements = (5-1)! = 4! = 24.
Couple sit together = (5-2)! * 2! = 3! * 2 = 6 * 2 = 12.
Couple do not sit together = (5-3)(5-2)! = 2 * 3! = 2 * 6 = 12.
The formula matches the calculation: (n-3)(n-2)! = (5-3)(5-2)! = 2 * 3! = 12.