Arranging People at a Round Table with Specific Conditions: A Comprehensive Analysis

Situation: Imagine a dinner party where 6 men and 6 women are to be seated around a round table. However, there's a specific condition: Bob, Ted, and Carol must sit together. How many different ways can they be seated to meet this condition?

The problem can be simplified by treating Bob, Ted, and Carol as a single entity or ldquo;block.rdquo; This approach reduces the complexity of the seating arrangement significantly. Let's break down the solution step-by-step.

Step 1: Treating the Block as a Single Unit

Why treat Bob, Ted, and Carol as a single unit? Because we need to ensure that they sit together, it's more convenient to consider them as a single unit. This block effectively reduces the number of entities we need to arrange from 12 to 10 (the block plus the remaining 9 individuals).

Step 2: Arranging the Units

Step 2.1: Arrangements of 10 Units
Arranging n people in a circular arrangement is done in (n-1)! ways. Therefore, for our 10 units, the number of circular arrangements is (10 - 1)! 9!

Step 2.2: Calculating the Factorial of 9
9! 362,880

Step 3: Arranging the Block

Within the block of 3 people (Bob, Ted, and Carol), they can be arranged in 3! ways.

Step 3.1: Calculating the Factorial of 3
3! 6

Step 4: Total Arrangements

The total number of arrangements is the product of the arrangements of the blocks and the arrangements within the block. Thus, the total number of ways is:

9! × 3! 362,880 × 6 2,177,280

Therefore, the total number of ways the 6 men and 6 women can sit at the round table with the condition that Bob, Ted, and Carol must sit together is 2,177,280.

Considering All Seating Arrangements

What about Alice? If we consider that Bob, Ted, and Alice can be seated in six different ways (BTC, BCT, CBT, CTB, TBC, TCB), it means that there are 10 ways for Bob, Ted, and Carol to be intermingled with the other nine people. Therefore, the total number of seating arrangements is:

6 × 362,880 21,772,800

Final Calculation with Fixed Seats

Let's assume the seats are numbered from 1 to 12, and Bob, Ted, and Carol are seated in seats 1 to 3. This means that any one of the nine remaining people can sit in seat 4, and so on. We can calculate the number of ways for the nine remaining people to be seated as:

9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 362,880

The same applies to the block of Bob, Ted, and Carol, giving us 6 ways. Therefore, the total number of combinations for each seating arrangement of the nine people is:

362,880 × 6 2,177,280

Considering that each person could move one seat around the table, this number of seating arrangements needs to be multiplied by 12 (the number of seats). This gives us:

2,177,280 × 12 26,127,360

In conclusion, if you want to predict which seat each diner would be sat at, knowing that Bob, Ted, and Carol were sat together, your chances of being correct are 1 in 26,127,360.