Question

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

Answer 1

36

Answer 2

Here's how to solve this problem:

1. Treat the men as a single unit: Since the 3 men must sit together, consider them as one entity.

2. Arrange the entities: Now you have 1 unit (the men) and 3 women, making a total of 4 entities to arrange around the circular table. The number of ways to arrange n distinct objects in a circle is (n-1)!. Therefore, there are (4-1)! = 3! = 6 ways to arrange these 4 entities.

3. Arrange the men within their unit: The 3 men can arrange themselves within their unit in 3! = 6 ways.

4. Multiply to find the total: The total number of arrangements is the product of these two results: 6 * 6 = 36.

Therefore, there are 36 ways to seat the 3 men and 3 women around the table with the men always sitting together.