5 men and 4 women are to be seated in a row so that the women occupy the even places . How many such arrangements are possible ?

Options

A. A. 2880
B. B. 1440
C. C. 720
D. D. 2020

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 1: Identify the total number of places
We have a total of 9 places in a row (1 to 9).

### Step 2: Identify the even places
The even places in the row are 2, 4, 6, and 8. This gives us a total of 4 even places where the women can sit.

### Step 3: Arrange the women in the even places
Since there are 4 women and 4 even places, we can arrange the women in these places. The number of ways to arrange 4 women is given by the factorial of the number of women, which is:

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 4: Arrange the men in the remaining places
After seating the women, we have 5 remaining places (1, 3, 5, 7, and 9) for the 5 men. The number of ways to arrange 5 men is given by the factorial of the number of men, which is:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements, we multiply the number of ways to arrange the women by the number of ways to arrange the men:

\[ \text{Total arrangements} = 4! \times 5! = 24 \times 120 = 2880 \]

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