In how many ways can 3 men and 3 women be seated around a round table such that all men are always together ?

Options

A. a. 88
B. b. 44
C. c. 72
D. d. 36

Answer

To solve the problem of seating 3 men and 3 women around a round table such that all men are always together, we can follow these steps:

### Step 1: Treat the group of men as a single unit
Since the 3 men must sit together, we can consider them as one single unit or block. Therefore, instead of arranging 6 individuals (3 men + 3 women), we will arrange 4 units: the block of men and the 3 individual women.

### Step 2: Calculate the arrangements of the units
When arranging objects around a round table, we use the formula for circular permutations, which is (n - 1)!, where n is the number of objects. Here, we have 4 units (the block of men + 3 women).

So, the number of ways to arrange these 4 units is:
\[ (4 - 1)! = 3! = 6 \]

### Step 3: Arrange the men within their block
Within the block of men, the 3 men can be arranged among themselves. The number of ways to arrange 3 men is given by:
\[ 3! = 6 \]

### Step 4: Combine the arrangements
Now, we need to multiply the number of arrangements of the units by the arrangements of the men within their block:
\[ \text{Total arrangements} = 3! \times 3! = 6 \times 6 = 36 \]

### Conclusion
Thus, the total number of ways to seat 3 men and 3 women around a round table such that all men are always together is **36**.