Answer these questions based on the following information.
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 seated around a circular table such that the men and women occupy the altenate positions?

Options

A. 288
B. 144
C. 72
D. 720

Answer

To solve the problem of seating 4 straight couples (4 men and 4 women) around a circular table such that men and women occupy alternate positions, we can follow these steps:

### Step 1: Fix the Position of One Woman
In a circular arrangement, we can fix one person's position to eliminate the effect of rotations. We can fix one woman in one seat. This leaves us with 3 remaining women to arrange.

**Hint:** When dealing with circular arrangements, fixing one position helps simplify the counting.

### Step 2: Arrange the Remaining Women
Now that one woman is fixed, we have 3 women left to arrange in the remaining 3 seats designated for women. The number of ways to arrange 3 women is given by \(3!\) (3 factorial).

\[ 3! = 3 \times 2 \times 1 = 6 \]

**Hint:** The factorial of a number \(n\) (denoted as \(n!\)) represents the number of ways to arrange \(n\) distinct objects.

### Step 3: Arrange the Men
Since men and women occupy alternate positions, after placing the women, there will be 4 seats left for the men. The number of ways to arrange the 4 men in these seats is given by \(4!\) (4 factorial).

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

**Hint:** Each arrangement of men is independent of how the women are arranged, so we can multiply the arrangements.

### Step 4: Calculate the Total Arrangements
To find the total number of arrangements, we multiply the number of ways to arrange the women by the number of ways to arrange the men:

\[ \text{Total arrangements} = 3! \times 4! = 6 \times 24 = 144 \]

**Hint:** When combining independent arrangements, multiply the number of ways each can occur.

### Conclusion
Thus, the total number of ways to seat the 4 couples around the circular table, with men and women occupying alternate positions, is **144**.

**Final Answer:** 144