There are five boys and three girls who are sitting together to discuss a management problem at a round table. In how many ways can they sit around the table so that no two girls are together?

Options

A. A) 1220
B. B) 1400
C. C) 1420
D. D) 1440

Answer

To solve the problem of arranging five boys and three girls around a round table such that no two girls sit next to each other, we can follow these steps:

### Step 1: Arrange the Boys
First, we need to arrange the five boys around the round table. The formula for arranging \( n \) items in a circular manner is \( (n-1)! \).

- Here, \( n = 5 \) (the number of boys).
- Therefore, the number of ways to arrange the boys is:
\[ (5-1)! = 4! = 24 \]

### Step 2: Identify the Spaces for Girls
Once the boys are arranged, there will be spaces available for the girls. In a circular arrangement of five boys, there will be five gaps (spaces) between them where the girls can sit.

### Step 3: Choose Spaces for the Girls
Next, we need to select spaces for the three girls from the five available spaces. We can use the combination formula \( \binom{n}{r} \) to choose \( r \) spaces from \( n \) available spaces.

- Here, we need to choose 3 spaces from the 5 available spaces:
\[ \binom{5}{3} = 10 \]

### Step 4: Arrange the Girls
After choosing the spaces for the girls, we need to arrange the three girls in the selected spaces. The number of ways to arrange \( r \) items is given by \( r! \).

- Here, we have 3 girls, so the number of ways to arrange them is:
\[ 3! = 6 \]

### Step 5: Calculate the Total Arrangements
Now, we can find the total number of arrangements by multiplying the number of ways to arrange the boys, the number of ways to choose the spaces for the girls, and the number of ways to arrange the girls.

- Total arrangements:
\[ \text{Total} = (\text{Ways to arrange boys}) \times (\text{Ways to choose spaces}) \times (\text{Ways to arrange girls}) \]
\[ \text{Total} = 24 \times 10 \times 6 = 1440 \]

### Final Answer
Thus, the total number of ways the boys and girls can sit around the table such that no two girls are together is **1440**.

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