In how many ways 6 persons can sit at a round table, if two of them prefer to sit together?

Answer

To solve the problem of how many ways 6 persons can sit at a round table if two of them prefer to sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 6 persons, and we need to arrange them in such a way that 2 specific persons (let's call them A and B) sit together.

2. **Treating A and B as One Unit**:
Since A and B want to sit together, we can treat them as a single unit or block. This means instead of arranging 6 individual persons, we will arrange 5 units: (AB), C, D, E, and F. Here, (AB) represents the block of A and B sitting together.

3. **Arranging the Units in a Circle**:
The formula for arranging n distinct objects in a circle is (n - 1)!. In our case, we have 5 units to arrange (the block (AB) and the other 4 persons). Therefore, the number of ways to arrange these 5 units in a circle is:
\[ (5 - 1)! = 4! = 24 \]

4. **Arranging A and B within Their Block**:
Within the block (AB), A and B can switch places. There are 2 ways to arrange A and B (either A sits first or B sits first). Thus, we multiply the arrangements of the units by the arrangements of A and B:
\[ 2! = 2 \]

5. **Calculating the Total Arrangements**:
Now, we multiply the number of arrangements of the 5 units by the arrangements of A and B:
\[ \text{Total arrangements} = 4! \times 2! = 24 \times 2 = 48 \]

### Final Answer:
Thus, the total number of ways in which 6 persons can sit at a round table, with A and B sitting together, is **48 ways**.

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