A group consists of four straight couple. That means each couple is having a male and a female.
In how many ways can they be arranged in a straight line such that no two men were sitting together?

Options

A. 1242
B. 1440
C. 3880
D. 2880

Answer

To solve the problem of arranging 4 straight couples (4 men and 4 women) in such a way that no two men sit together, we can follow these steps:

### Step 1: Arrange the Women
First, we will arrange the 4 women. The number of ways to arrange 4 women is given by the factorial of the number of women:

\[ \text{Ways to arrange women} = 4! = 24 \]

### Step 2: Identify the Positions for Men
Once the women are arranged, we need to determine where the men can sit. The arrangement of 4 women creates 5 possible positions for the men:

1. Before the first woman
2. Between the first and second women
3. Between the second and third women
4. Between the third and fourth women
5. After the fourth woman

This gives us a total of 5 positions.

### Step 3: Choose Positions for Men
Since we need to place 4 men in these 5 available positions, we can choose 4 positions out of the 5. The number of ways to choose 4 positions from 5 is given by the combination formula:

\[ \text{Ways to choose positions} = \binom{5}{4} = 5 \]

### Step 4: Arrange the Men
After choosing the positions for the men, we can arrange the 4 men in those selected positions. The number of ways to arrange 4 men is also given by the factorial of the number of men:

\[ \text{Ways to arrange men} = 4! = 24 \]

### Step 5: Calculate the Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the women, the number of ways to choose positions for the men, and the number of ways to arrange the men:

\[ \text{Total arrangements} = (4!) \times \binom{5}{4} \times (4!) \]
\[ = 24 \times 5 \times 24 \]
\[ = 2880 \]

Thus, the total number of ways to arrange the couples such that no two men are sitting together is **2880**.

---