There are 5 gentlemen and 4 ladies to dine at a round table. In how many ways can they as themselves so that no that ladies are together ?

Options

A. 2880
B. 576
C. 1440
D. 480

Explanation

To solve the problem of arranging 5 gentlemen and 4 ladies at a round table such that no two ladies are sitting together, we can follow these steps:

### Step 1: Arrange the Gentlemen
First, we need to arrange the 5 gentlemen around the round table. The formula for arranging \( n \) objects in a circle is given by \( (n-1)! \). Therefore, for 5 gentlemen, the number of ways to arrange them is:

\[ (5 - 1)! = 4! = 24 \]

### Step 2: Identify Spaces for Ladies
Once the gentlemen are seated, we can identify the spaces available for the ladies. When 5 gentlemen are seated, they create 5 gaps (spaces) between them where the ladies can sit. These gaps are:

1. Between Gentleman 1 and Gentleman 2
2. Between Gentleman 2 and Gentleman 3
3. Between Gentleman 3 and Gentleman 4
4. Between Gentleman 4 and Gentleman 5
5. Between Gentleman 5 and Gentleman 1

### Step 3: Choose Spaces for the Ladies
Since we have 5 gaps and we need to choose 4 of these gaps to place the ladies, we can use the combination formula \( \binom{n}{r} \) which gives us the number of ways to choose \( r \) items from \( n \) items without regard to the order of selection. Thus, the number of ways to choose 4 gaps from 5 is:

\[ \binom{5}{4} = 5 \]

### Step 4: Arrange the Ladies
After choosing the 4 gaps, we need to arrange the 4 ladies in these selected gaps. The number of ways to arrange 4 ladies is given by \( 4! \):

\[ 4! = 24 \]

### Step 5: Calculate the Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the gentlemen, the number of ways to choose the gaps, and the number of ways to arrange the ladies:

\[ \text{Total arrangements} = (4!) \times \binom{5}{4} \times (4!) = 24 \times 5 \times 24 \]

Calculating this gives:

\[ 24 \times 5 = 120 \]
\[ 120 \times 24 = 2880 \]

Thus, the total number of ways to arrange 5 gentlemen and 4 ladies at a round table such that no two ladies are together is **2880**.

### Final Answer
The total number of arrangements is **2880**.

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