Question

There are five people: \(A, B, C, D,\) and \(E\) The following pairs know each other: \(\mathrm{A}\) and \(\mathrm{C}, \mathrm{B}\) and \(\mathrm{C}, \mathrm{A}\) and \(\mathrm{D}, \mathrm{D}\) and \(\mathrm{E},\) and C and D. a) Arrange the five people in a row so that nobody is next to a stranger. b) How many different arrangements are possible such that nobody is next to a stranger? c) The five people are joined by a sixth person, \(F\), who knows only A. In how many ways can the six people stand in a row if nobody can be next to a stranger? Explain your answer.

Short answer

a) A, C, B, D, Eb) 8 arrangementsc) 24 arrangements

Step-by-step solution

Identify connections

List all connections: A knows C and D, B knows C, C knows A, B, and D, D knows C, A, and E, and E knows D.

Draw a graph

Draw a graph representing the connections among the people, with vertices as people and edges as the relationships.

Form groups with mutual acquaintances

Group the people based on their mutual acquaintances: {A, C, D}, {B, C}, {D, E}.

Determine a valid arrangement (Part a)

Determine a linear arrangement where everyone is next to a known person. Example: A, C, B, D, E.

Count valid arrangements using combinatorics (Part b)

Count how many such arrangements are possible. Each group can rearrange within itself. For our groups {A, C, D} and {D, E}, we have permutations: A, C, D/E or E, D.

Adding the 6th person (Part c)

Now include F, who only knows A. Thus A and F must be together. We have {(A,F), C, B, D, E}

Count combinations for 6 people

Now count all permutations with (A,F) fixed positions. For F fixed, group (C, B, D, E) permutations remains.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects according to specified rules. In permutation problems, we deal with arranging objects in specific sequences. The permutations have strict rules for positioning, where the order of objects greatly matters.
For our exercise, consider arranging people in such a way that no one stands next to a stranger. A stranger is someone they do not have a direct acquaintance with. Therefore, the combinatorial focus will be on finding valid sequences or permutations that adhere to these acquaintance rules.
By understanding combinatorics, students can grasp the importance of each position in the sequence and how the relationships between elements (people) influence the number of valid permutations.

Graph Theory

Graph theory is a field in mathematics focusing on the study of graphs, which are mathematical structures used to model pairwise relations between objects. In this exercise, each person can be considered a vertex, and each acquaintance relationship is represented by an edge connecting two vertices.
In our problem, the graph helps visualize the connections between people:

• Vertices: A, B, C, D, E.
• Edges: (A, C), (B, C), (A, D), (D, E), (C, D).

Drawing this graph allows us to easily see who knows whom and assists in forming groups where everyone is connected either directly or indirectly. For instance, we identify the groups like {A, C, D}, {B, C}, and {D, E} easier. This visual representation simplifies understanding the relationships necessary for valid permutations.

Mutual Acquaintances

Mutual acquaintances mean that if person A knows person B, then person B also knows person A. This mutual relationship is crucial for arranging people so that no one is next to a stranger. In our problem, determining groups of mutual acquaintances ensures that everyone within a group knows each other directly or through another member:

• Group 1: A, C, D - Each person either directly knows the others or is connected through another person.
• Group 2: B, C - B knows C and vice versa.
• Group 3: D, E - They are directly connected.

Identifying these groups through mutual acquaintances allows us to effectively break down the problem and ensure no strangers are next to each other in the final arrangement. This makes some permutations possible while others are not.

Permutation Counting

Permutation counting involves calculating the number of valid sequences where objects are arranged based on certain conditions. Analyzing our problem, we need to calculate permutations where each person is next to someone they know.
For part (b), we count valid arrangements by organizing people into permissible groups and then calculating the permutations within these groups:

• {A, C, D}
• {B, C}
• {D, E}

Each group's internal arrangement can vary, contributing to the total number of permutations. Additionally, mutual acquaintances reduce overall complexity, as fewer unique arrangements are valid.
For part (c), introducing a new person F who only knows A adds complexity. F and A must stand together, turning (A, F) into a single unit. This combination reduces permutation possibilities but remains calculable by fixing F's place and permuting the remaining group members. Thus, permutation counting combines combinatorial principles with constraints posed by mutual acquaintances to tally valid sequences efficiently.