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

Answer

To solve the problem of how many ways 5 gentlemen and 4 ladies can seat themselves at a round table such that no two ladies are together, we can follow these steps:

### Step 1: Arrange the gentlemen
Since the seating is at a round table, we can fix one gentleman to eliminate the effect of rotations. The remaining 4 gentlemen can be arranged in the remaining seats.

- The number of ways to arrange 4 gentlemen is given by (n-1)! where n is the number of gentlemen.

\[ \text{Number of ways to arrange gentlemen} = (5-1)! = 4! = 24 \]

### Step 2: Identify positions for the ladies
After seating the 5 gentlemen, we need to find the positions available for the ladies. When the gentlemen are seated, they create gaps between them where the ladies can sit.

- With 5 gentlemen seated, there are 5 gaps (one between each pair of gentlemen) where the ladies can be seated.

### Step 3: Choose positions for the ladies
We need to choose 4 out of these 5 gaps to place the ladies, ensuring that no two ladies sit together.

- The number of ways to choose 4 gaps from 5 is given by the combination formula \( \binom{n}{r} \):

\[ \text{Number of ways to choose positions for ladies} = \binom{5}{4} = 5 \]

### Step 4: Arrange the ladies in the chosen positions
Once the positions are chosen, we can arrange the 4 ladies in those positions.

- The number of ways to arrange 4 ladies is given by 4!:

\[ \text{Number of ways to arrange ladies} = 4! = 24 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements where no two ladies are together, we multiply the number of arrangements of gentlemen, the number of ways to choose the positions for the ladies, and the number of arrangements of the ladies:

\[ \text{Total arrangements} = (\text{Ways to arrange gentlemen}) \times (\text{Ways to choose positions}) \times (\text{Ways to arrange ladies}) \]

\[ \text{Total arrangements} = 24 \times 5 \times 24 = 2880 \]

### Final Answer
Thus, the total number of ways the 5 gentlemen and 4 ladies can seat themselves at a round table such that no two ladies are together is **2880**.

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