In how many ways can 5 men and 2 ladies be arranged at a round table if two ladies are never together?

Options

A. 5040
B. 480
C. 240
D. 720

Answer

To solve the problem of arranging 5 men and 2 ladies at a round table such that the two ladies are never seated next to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Arrange the Men:**
- Since the arrangement is circular, we can fix one man to eliminate the effect of rotation. Therefore, we arrange the remaining 4 men around him.
- The number of ways to arrange 5 men in a circular manner is given by \((n-1)!\), where \(n\) is the number of men.
- Thus, the number of arrangements for the men is:
\[ 4! = 24 \]

2. **Identify Gaps for Ladies:**
- After arranging the 5 men, they create gaps where the ladies can be seated. In a circular arrangement of 5 men, there are 5 gaps (one between each pair of men).
- These gaps can be visualized as follows:
- Gap 1 (between Man 1 and Man 2)
- Gap 2 (between Man 2 and Man 3)
- Gap 3 (between Man 3 and Man 4)
- Gap 4 (between Man 4 and Man 5)
- Gap 5 (between Man 5 and Man 1)

3. **Choose Gaps for the Ladies:**
- We need to select 2 out of these 5 gaps to place the ladies. The number of ways to choose 2 gaps from 5 is given by the combination formula \(\binom{n}{r}\), where \(n\) is the total number of gaps and \(r\) is the number of gaps to choose.
- Therefore, the number of ways to choose 2 gaps from 5 is:
\[ \binom{5}{2} = 10 \]

4. **Arrange the Ladies:**
- The two ladies can be arranged in the selected gaps in \(2!\) ways (since there are 2 ladies).
- Thus, the number of arrangements for the ladies is:
\[ 2! = 2 \]

5. **Calculate the Total Arrangements:**
- Now, we multiply the number of arrangements of the men, the ways to choose the gaps, and the arrangements of the ladies:
\[ \text{Total arrangements} = 4! \times \binom{5}{2} \times 2! = 24 \times 10 \times 2 \]
- Calculating this gives:
\[ 24 \times 10 = 240 \]
\[ 240 \times 2 = 480 \]

### Final Answer:
The total number of ways to arrange 5 men and 2 ladies at a round table such that the two ladies are never seated next to each other is **480**.