(i) Let D1 and D2 be the two particular delegates. Considering D1 and D2 as one delegate, we have 19 delegates in all. 19 delegates can be seated round a circular table in (19 – 1)! = 18 ! ways.
But two particular delegates can seat themselves in 2! (D1 D2 or D2 D1) ways.
Hence, the total number of ways = 18! × 2! = 2 (18!)
(ii) To find the number of ways in which two particular delegates never sit together, we subtract the number of ways in which they sit together from the total number of ways of seating 20 persons i.e., (20 – 1)! = 19! ways.
Hence the total number of ways in this case = 19! – 2 (18!) = 19 (18!) – 2 (18!) = 17 (18!).