A round-table conference is to be held among 20 delegates belonging from 20 different countries. In how many ways can they be seated if two particular delegates are (i) always to sit together, (ii) never to sit together .

Explanation

(i) Let the two particular delegates who wish to sit together be treated as one delegate. So we have 19 delegates who can be arranged on a round table in (19-1)!, i.e., 18! Ways.
After this, the two particular delegates can be permuted between themselves in 2!=2 ways. Hence, by product rule, number of required arrangements is `2xx(18)!`.
(ii) The total number of arrangements of 20 delegates on a round table is 19!.
Hence, the number of arrangements in which the two particular delegates never sit together is `19!-2xx18!=18! (19-2)=17xx18!`.