Number of ways in which `5` boys and `4` girls can be arranged on a circular table such that no two girls sit together and two particular boys are always together: (A) 276 (B) 288 (C) 296 (D) 304

Options

A. 288
B. 44
C. 720
D. 540

Answer

To solve the problem of arranging `5` boys and `4` girls on a circular table such that no two girls sit together and two particular boys are always together, we can follow these steps:

### Step-by-Step Solution:

1. **Treat the two particular boys as one unit**: Since two particular boys are always together, we can consider them as a single unit or block. This reduces the number of boys from `5` to `4` (the block of 2 boys + 3 other boys).

2. **Count the total units**: After treating the two boys as one unit, we have `4` boys (the block + 3 other boys) and `4` girls. This gives us a total of `4 boys + 4 girls = 8 units` to arrange.

3. **Arrange the boys**: When arranging `n` units in a circular manner, the number of arrangements is given by `(n-1)!`. Here, we have `4` boys (the block counts as one unit), so the number of ways to arrange the boys is `(4 - 1)! = 3!`.

4. **Arrange the two particular boys within their block**: The two particular boys can be arranged among themselves in `2!` ways.

5. **Arrange the girls**: Since no two girls can sit together, we can place the girls in the gaps created by the boys. With `4` boys arranged, there are `4` gaps available for the girls. The girls can be arranged in these gaps in `4!` ways.

6. **Calculate the total arrangements**: The total number of arrangements is the product of the arrangements of the boys, the arrangements of the two boys within their block, and the arrangements of the girls:
\[ \text{Total arrangements} = (3!) \times (2!) \times (4!) \]

7. **Calculate the factorial values**:
- \(3! = 6\)
- \(2! = 2\)
- \(4! = 24\)

8. **Final Calculation**:
\[ \text{Total arrangements} = 6 \times 2 \times 24 = 288 \]

Thus, the total number of ways to arrange `5` boys and `4` girls on a circular table such that no two girls sit together and two particular boys are always together is **288**.