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Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group?

Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title:** Friendship Possibilities in a Group **Text:** Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group? --- **Explanation:** The problem explores social connections within a set number of people. The question is whether each individual can have a specified number of friendships within a group of five. **Graphical Representation:** To visualize this problem, you can use a diagram where each person is represented by a node, and each friendship by a line connecting two nodes. 1. **For 2 Friends Each:** - Try forming a cycle where each person (node) is connected to two others in a closed loop. - Imagine the layout as a pentagon where each vertex connects to the two nearest vertices. 2. **For 3 Friends Each:** - A possible approach is connecting each person to non-adjacent nodes. - This can be visualized as a complete graph minus two edges, ensuring all nodes maintain three connections. Through these visual models, we can explore if all conditions can be satisfactorily met within the constraints of the problem.
Transcribed Image Text:**Title:** Friendship Possibilities in a Group **Text:** Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group? --- **Explanation:** The problem explores social connections within a set number of people. The question is whether each individual can have a specified number of friendships within a group of five. **Graphical Representation:** To visualize this problem, you can use a diagram where each person is represented by a node, and each friendship by a line connecting two nodes. 1. **For 2 Friends Each:** - Try forming a cycle where each person (node) is connected to two others in a closed loop. - Imagine the layout as a pentagon where each vertex connects to the two nearest vertices. 2. **For 3 Friends Each:** - A possible approach is connecting each person to non-adjacent nodes. - This can be visualized as a complete graph minus two edges, ensuring all nodes maintain three connections. Through these visual models, we can explore if all conditions can be satisfactorily met within the constraints of the problem.
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