The number of ways in which 5 boys and 4 girls sit around a circular table so that no two girls sit together is:

Options

A. a. 3600
B. b. 120o
C. c. 2880
D. d. 1320

Answer

To find the number of ways in which 5 boys and 4 girls can sit around a circular table such that no two girls sit together, we can follow these steps:

### Step 1: Arrange the Boys
Since the arrangement is circular, we can fix one boy to eliminate the effect of rotations. Therefore, we arrange the remaining 4 boys around the table.

- The number of ways to arrange 5 boys in a circular manner is given by (n-1)!, where n is the number of boys.
- Here, n = 5, so we have:
\[ \text{Ways to arrange boys} = (5-1)! = 4! = 24 \]

### Step 2: Identify Spaces for Girls
Once the boys are arranged, they create spaces where the girls can sit. With 5 boys, there will be 5 gaps between them (including the gap before the first boy and after the last boy).

- The arrangement looks like this: B _ B _ B _ B _ B
- There are 5 gaps available for the girls.

### Step 3: Place the Girls
We need to place 4 girls in these 5 gaps such that no two girls sit together. This means we can choose any 4 out of the 5 available gaps.

- The number of ways to choose 4 gaps from 5 is given by the combination formula:
\[ \binom{5}{4} = 5 \]

### Step 4: Arrange the Girls
After choosing the gaps, we can arrange the 4 girls in those selected gaps. Since the girls are distinct, the number of arrangements of the 4 girls is:
\[ 4! = 24 \]

### Step 5: Calculate the Total Arrangements
Finally, we multiply the number of arrangements of boys, the number of ways to choose the gaps, and the arrangements of girls to get the total number of arrangements:
\[ \text{Total arrangements} = \text{Ways to arrange boys} \times \text{Ways to choose gaps} \times \text{Ways to arrange girls} \]
\[ = 24 \times 5 \times 24 = 2880 \]

### Final Answer
The total number of ways in which 5 boys and 4 girls can sit around a circular table such that no two girls sit together is **2880**.

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