In how many ways can 7 boys be seated at a round table so that two particular boys are next to each other

Answer

To find the number of ways to seat 7 boys at a round table such that two particular boys are next to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Treat the Two Particular Boys as One Unit**:
Since we want the two particular boys (let's call them A and B) to sit next to each other, we can treat them as a single unit or block. Thus, instead of 7 boys, we now have 6 units to arrange: the block (A and B) and the other 5 boys.

2. **Arrange the Units in a Circle**:
The number of ways to arrange n units in a circle is given by (n-1)!. Here, we have 6 units (the block and the 5 other boys), so we will arrange them in (6-1)! = 5! ways.

3. **Calculate 5!**:
5! = 5 × 4 × 3 × 2 × 1 = 120.

4. **Arrange A and B within Their Block**:
Since A and B can switch places within their block, we have 2 ways to arrange them (A next to B or B next to A).

5. **Multiply the Arrangements**:
To find the total number of arrangements, we multiply the number of ways to arrange the 6 units by the number of arrangements of A and B within their block:
Total arrangements = 5! × 2! = 120 × 2 = 240.

### Final Answer:
Thus, the total number of ways to seat 7 boys at a round table such that two particular boys are next to each other is **240**.