Final answer:
For 15 people to each have exactly 3 friends, it is not possible as you would end up needing a half-edge, which is impossible in graph theory. For 4 people each having 3 friends, it is possible, represented by a complete graph (K4) in graph theory.
Step-by-step explanation:
The problem asked is related to an application of the Handshake Theorem in graph theory, which can be applied to solve problems regarding friendships within a group of people, where friendships are mutual.
Considering a group of 15 people, where each person has exactly 3 friends, we can represent this scenario using a graph where the vertices correspond to people and the edges correspond to friendships between them. According to the Handshake Theorem, which states that the sum of the degrees of all vertices in a graph is twice the number of edges, we can see that for each person to have 3 friends, the total number of handshakes is 3 times 15, divided by 2 (since each handshake is counted twice), giving us 22.5, which is not possible since we cannot have a half-edge. Therefore, it is impossible for each person in a group of 15 people to have exactly 3 friends.
In a group of 4 people, it is possible for each person to have exactly 3 friends. Again, representing this with a graph, the vertices are the people and the edges are the friendships between them. Since the Handshake Theorem would require 3 times 4, divided by 2 edges, which equals 6, it is possible to have a graph where each vertex is connected to every other vertex, forming a complete graph (K4). It means every person can indeed have three friends within this group.