Four men and three women are to be seated for a dinner such that no two women sit together and no two sit together and no two men sit together . Find the number of ways in which this can be arranged .

Options

A. A. 144
B. B. 72
C. C. 36
D. D. none of these

Answer

To solve the problem of seating four men and three women at a dinner such that no two women sit together and no two men sit together, we can follow these steps:

### Step 1: Arranging the Men
First, we will arrange the four men. Since there are four men, they can be arranged in \(4!\) (factorial of 4) ways.

\[ 4! = 24 \text{ ways} \]

### Step 2: Identifying the Seats for Women
Once the men are seated, we need to identify the available seats for the women. When four men are seated, they create five potential gaps for the women to sit: one before the first man, one between each pair of men, and one after the last man. This can be visualized as:

- _ M _ M _ M _ M _

Here, "M" represents a man, and "_" represents a possible seat for a woman. Therefore, there are 5 gaps available.

### Step 3: Arranging the Women
Next, we need to choose 3 out of these 5 gaps to seat the women. The number of ways to choose 3 gaps from 5 is given by the combination formula \(C(n, r)\), where \(n\) is the total number of gaps and \(r\) is the number of gaps to choose.

\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \text{ ways} \]

After choosing the gaps, we can arrange the three women in those selected gaps. The three women can be arranged in \(3!\) ways.

\[ 3! = 6 \text{ ways} \]

### Step 4: Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the men, the number of ways to choose the gaps for the women, and the number of ways to arrange the women.

\[ \text{Total arrangements} = (4!) \times C(5, 3) \times (3!) \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 24 \times 10 \times 6 = 1440 \]

Thus, the total number of ways to arrange four men and three women such that no two women sit together and no two men sit together is **1440**.