4 men and 4 women are to be seated for a dinner, In a row, such that men and women sit alternately. Find the number of ways in which this arrangements can be done.

Options

A. 1152
B. 1252
C. 576
D. 40320

Answer

To solve the problem of arranging 4 men and 4 women alternately in a row, we can follow these steps:

### Step 1: Determine the Arrangement Patterns
Since the arrangement must be alternating, there are two possible patterns:
1. Man - Woman - Man - Woman - Man - Woman - Man - Woman
2. Woman - Man - Woman - Man - Woman - Man - Woman - Man

### Step 2: Calculate the Arrangements for Each Pattern
For each of the two patterns, we need to arrange 4 men and 4 women.

- **Arranging Men**: The number of ways to arrange 4 men is given by the factorial of the number of men:
\[ \text{Ways to arrange men} = 4! = 24 \]

- **Arranging Women**: Similarly, the number of ways to arrange 4 women is:
\[ \text{Ways to arrange women} = 4! = 24 \]

### Step 3: Combine the Arrangements
Since there are two patterns (starting with a man or starting with a woman), we multiply the arrangements by 2:
\[ \text{Total arrangements} = 2 \times (4! \times 4!) = 2 \times (24 \times 24) \]

### Step 4: Calculate the Final Result
Now we compute the total number of arrangements:
\[ \text{Total arrangements} = 2 \times (24 \times 24) = 2 \times 576 = 1152 \]

Thus, the total number of ways in which the arrangements can be done is **1152**.