In how many ways could 12 people be seated at a rectangular table if the two hosts must not sit together?
To determine the number of ways to seat 12 people at a rectangular table where the two hosts must not sit together, we can break down the problem into smaller steps.
Step 1: Determine the total number of ways to seat 12 people without any restrictions.
The first person can be seated at any of the 12 available seats. After that, the second person can be seated in any of the remaining 11 seats, and so on. Therefore, the total number of ways to seat 12 people in a rectangular table without any restrictions is given by 12 factorial (12!).
Step 2: Determine the number of ways two hosts can sit together.
Since the two hosts must sit together, we can treat them as a single unit. This means we consider the two hosts as a single entity that needs to be seated at a rectangular table with 11 available seats. The two hosts can be arranged in 2 factorial (2!) ways. Additionally, the remaining 11 people can be seated in 11 factorial (11!) ways. Therefore, the total number of ways the two hosts can sit together is given by 2! * 11!.
Step 3: Determine the number of ways the two hosts must not sit together.
To find the number of ways the two hosts must not sit together, we subtract the number of ways the two hosts can sit together from the total number of ways without any restrictions. So, the number of ways the two hosts must not sit together is given by 12! - (2! * 11!).
To obtain the final answer, calculate the value of 12! - (2! * 11!).