Answer these questions independently of each other
5 men and 4 women are to be seated for a dinner, In a row, such that women occupy the even places. Find the number of ways in which this arrangements can be done.

Options

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

Answer

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

### Step 1: Identify the seating positions
In a row of 9 seats, the positions are numbered from 1 to 9. The even positions are 2, 4, 6, and 8. This means that we have 4 even positions available for the 4 women.

### Step 2: Arrange the women in the even positions
Since we have 4 women and 4 even positions, we can arrange the 4 women in these positions. The number of ways to arrange 4 women is given by the factorial of the number of women, which is \(4!\).

\[ 4! = 24 \]

### Step 3: Arrange the men in the remaining positions
After placing the women in the even positions, we have 5 men left to arrange in the remaining 5 positions (1, 3, 5, 7, and 9). The number of ways to arrange 5 men is given by \(5!\).

\[ 5! = 120 \]

### Step 4: 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.

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

### Final Answer
The total number of ways in which the arrangements can be done is **2880**.

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