A round table coference is to be held between 20 delegates. How many seating arrangements are possible if two particular delegates are always to sit together .

Answer

To solve the problem of arranging 20 delegates at a round table where two particular delegates must sit together, we can follow these steps:

### Step 1: Treat the two delegates as a single unit
Since two delegates (let's call them A and B) must sit together, we can treat them as one single unit or block. This means instead of arranging 20 individual delegates, we now have 19 units to arrange (the block of A and B plus the other 18 delegates).

### Step 2: Calculate the circular arrangements of the units
In a circular arrangement, the number of ways to arrange \( n \) units is given by \( (n - 1)! \). Here, we have 19 units, so the number of circular arrangements is:
\[ (19 - 1)! = 18! \]

### Step 3: Account for the arrangement within the block
Within the block of A and B, these two delegates can switch places. Therefore, there are 2 ways to arrange A and B within their block.

### Step 4: Combine the arrangements
To find the total number of arrangements, we multiply the number of arrangements of the 19 units by the arrangements of A and B within their block:
\[ \text{Total arrangements} = 18! \times 2 \]

### Final Answer
Thus, the total number of seating arrangements possible is:
\[ 2 \times 18! \]