Question

A party has \(n\) guests. Two of the guests do not get along well with each other. In how many ways can the guests be seated in a row so that these two persons do not sit next to each other?

Short answer

The number of ways is \( n! - 2(n-1)! \).

Step-by-step solution

Calculate Total Arrangements

First, calculate the total number of ways to arrange the guests without any restrictions. Since there are \( n \) guests, we can arrange them in \( n! \) different ways.

Calculate Arrangements With Restrictions

Imagine the two guests who don't get along as one "block" of two people sitting together. This block, plus the remaining \( n-2 \) guests, makes \( n-1 \) units to arrange along the row. The number of ways to arrange these "units" is \( (n-1)! \). Since the block can internally be arranged in 2 ways itself (guest A on the left of guest B, or guest B on the left of guest A), multiply by 2. The total restricted arrangements, where they sit next to each other, is: \( 2 \times (n-1)! \).

Subtract to Find Desired Arrangements

Subtract the arrangements where the two guests sit together (from Step 2) from the total arrangements (from Step 1) to find the total number of arrangements where they do not sit together: \( n! - 2 \times (n-1)! \).

Key concepts

Permutations

Permutations refer to the different ways in which a set of items can be arranged or ordered. In the context of our problem, permutations help us calculate how many ways we can organize the seatings of guests at a party. When we talk about permutations of "n" distinct objects, we're interested in all the possible sequences we can form where each arrangement is unique.

• For example, with 3 guests, the different permutations or orderings of the guests might be ABC, ACB, BAC, BCA, CAB, and CBA, totaling 6 permutations.
• This is calculated using the formula for permutations: \( n! \), which reads as "n factorial." Here, "n" represents the number of objects we are arranging.

Understanding permutations is essential for figuring out not just how to place guests, but also how many valid arrangements follow certain rules or restrictions.

Factorials

Factorials are a powerful mathematical operation essential when dealing with permutations and combinations. A factorial, designated with an exclamation point (!), is the product of an integer and all the integers below it down to 1. So, \( n! \) is the product \( n \times (n-1) \times (n-2) \times \, \ldots \, \times 1 \).

• For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Factorials provide us with the total number of possible permutations for n distinct items.

In seating arrangement problems, factorials help us determine how many unique ways we can arrange guests when no seating restrictions are applied. This forms the baseline from which we subtract restricted cases.

Restrictive Conditions

When certain conditions are applied, some permutations are invalid or unwanted. In our exercise, two guests must not sit together. These conditions create constraints that limit the number of permissible arrangements.

• Initially, we calculate the total arrangements without restrictions.
• Next, identify and count the unwanted arrangements—that is, where the two guests do sit together.
• Consider "blocking" the two guests who do not get along into one unit when they are sitting together.
• Calculate the permutations as if this block were a single unit or, effectively, one guest.

By subtracting these restricted permutations from the unrestricted total, we hone in on the valid seating arrangements that respect the condition.

Seating Arrangements

Seating arrangements involve determining how to place people in a particular order, which is often complicated by additional factors like rules or restrictions. This classical problem ties together permutations, factorials, and restrictive conditions.

• First, determine the total possible arrangements when guests are seated without any restrictions using factorials.
• Introduce restrictions, such as specific guests not sitting together, which affect the calculation of valid arrangements.
• By grouping restricted guests as a block and subtracting the permutations of this block, you identify arrangements where all seating conditions are satisfied.

Understanding seating arrangements with constraints teaches valuable problem-solving skills in combinatorics, helping solve real-world logistical issues efficiently.