It is required to seat 5 men and 3 women in a row so that the women occupy the even places. How many such arrangements are possible ?

Explanation

To solve the problem of seating 5 men and 3 women in a row such that the women occupy the even places, we can follow these steps:

### Step 1: Identify the positions
In a row of 8 seats (5 men + 3 women), the even positions are 2, 4, 6, and 8. Therefore, there are 4 even positions available for the 3 women.

### Step 2: Choose positions for the women
We need to select 3 out of the 4 available even positions for the women. The number of ways to choose 3 positions from 4 is given by the combination formula:

\[ \text{Number of ways to choose positions} = \binom{4}{3} = 4 \]

### Step 3: Arrange the women in the chosen positions
Once we have chosen the positions, we can arrange the 3 women in those positions. The number of ways to arrange 3 women is given by the factorial of the number of women:

\[ \text{Number of arrangements of women} = 3! = 6 \]

### Step 4: Arrange the men
After placing the women, we have 5 positions left for the men. The number of ways to arrange the 5 men in these positions is given by:

\[ \text{Number of arrangements of men} = 5! = 120 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements, we multiply the number of ways to choose the positions for the women, the arrangements of the women, and the arrangements of the men:

\[ \text{Total arrangements} = \text{(Ways to choose positions)} \times \text{(Arrangements of women)} \times \text{(Arrangements of men)} \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 4 \times 6 \times 120 = 2880 \]

### Final Answer
Thus, the total number of arrangements in which 5 men and 3 women can be seated in a row such that the women occupy the even places is **2880**.

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