It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Options

A. 2880
B. 2480
C. 3680
D. 3280

Answer

To solve the problem of seating 5 men and 4 women in a row such that the women occupy the even places, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Seats**:
We have a total of 9 seats (5 men + 4 women = 9).

2. **Determine the Even Places**:
The even places in a row of 9 seats are positions 2, 4, 6, and 8. Thus, there are 4 even positions available for the women.

3. **Arrange the Women**:
Since there are 4 women and they will occupy the 4 even places, we can arrange the 4 women in these 4 positions. The number of ways to arrange 4 women is given by the factorial of the number of women:
\[ \text{Ways to arrange women} = 4! = 24 \]

4. **Arrange the Men**:
The remaining positions (1, 3, 5, 7, and 9) will be occupied by the 5 men. The number of ways to arrange 5 men in these 5 positions is given by the factorial of the number of men:
\[ \text{Ways to arrange men} = 5! = 120 \]

5. **Calculate Total Arrangements**:
Since the arrangements of women and men are independent, we multiply the number of arrangements of women by the number of arrangements of men to get the total arrangements:
\[ \text{Total arrangements} = \text{Ways to arrange women} \times \text{Ways to arrange men} = 24 \times 120 = 2880 \]

Thus, the total number of arrangements where 5 men and 4 women can be seated such that women occupy the even places is **2880**.