5 men and 6 women have to be seated in a straight row so that no two women are together. Find the number of ways this can be done.

Options

A. 48400
B. 39600
C. 9900
D. 86400

Answer

To solve the problem of seating 5 men and 6 women in a straight row such that no two women are together, we can follow these steps:

### Step 1: Arrange the Men
First, we will arrange the 5 men. The number of ways to arrange 5 men is given by the factorial of the number of men:

\[ \text{Ways to arrange men} = 5! = 120 \]

### Step 2: Identify Positions for the Women
Once the men are arranged, we can visualize the arrangement. The arrangement of 5 men creates 6 possible gaps (positions) where the women can be seated. These gaps are:
- Before the first man
- Between the first and second man
- Between the second and third man
- Between the third and fourth man
- Between the fourth and fifth man
- After the fifth man

This can be illustrated as follows:
```
_ M1 _ M2 _ M3 _ M4 _ M5 _
```
Here, each underscore represents a potential position for a woman.

### Step 3: Choose Positions for the Women
Since we need to place 6 women in these 6 gaps and no two women can be together, we must place one woman in each gap. Thus, we will fill all 6 gaps with the 6 women.

### Step 4: Arrange the Women
The number of ways to arrange the 6 women in the 6 positions is given by the factorial of the number of women:

\[ \text{Ways to arrange women} = 6! = 720 \]

### Step 5: Calculate the Total Arrangements
To find the total number of arrangements of 5 men and 6 women such that no two women are together, we multiply the number of arrangements of men by the number of arrangements of women:

\[ \text{Total arrangements} = (5!) \times (6!) = 120 \times 720 = 86400 \]

Thus, the total number of ways to arrange 5 men and 6 women in a straight row such that no two women are together is **86400**.

### Final Answer
The number of ways to seat 5 men and 6 women in a straight row so that no two women are together is **86400**.

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