Six people, including Kavi and Lata, are to be seated around a round table. Two seatings are taken to be the same if one can be turned into the other just by everyone moving round the table together. In how many ways can they be seated so that Kavi and Lata never sit next to each other?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 72
Around a round table only the order of the people matters, not where the circle starts, so \(n\) people can be seated in \((n-1)!\) ways. "Never next to each other" is found as all seatings minus the seatings with the two together.
All seatings of 6 people: \((6-1)!=5!=120\)
Kavi and Lata together: tie them into one unit; 5 units round the table give \((5-1)!=4!=24\) seatings
Inside the unit they can sit as Kavi-Lata or Lata-Kavi: \(2!=2\) ways
Seatings with the two together \(=24\times2=48\)
Seatings with the two never together \(=120-48=72\)
Why the other options are incorrect
Option (1) 48 — Incorrect. This is the number of seatings in which they do sit together.
Option (2) 120 — Incorrect. This is all the seatings, including the 48 in which Kavi and Lata sit side by side.
Option (4) 96 — Incorrect. This is \(120-24\); it forgets that the two can sit inside their unit in 2 orders.