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

To solve the problem, we will break it down into two parts as specified:

### Part (i): Two particular delegates always sit together.

1. **Consider the two delegates as a single unit**: Since the two particular delegates must sit together, we can treat them as one unit or "block". This means instead of 20 delegates, we now have 19 units to arrange (18 individual delegates + 1 block of 2 delegates).

2. **Calculate the arrangements of these units**: For a round table arrangement, the number of ways to arrange n units is given by (n-1)!. Here, n = 19, so we have:
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