Final answer:
In graph theory, the question asks if a 3-regular graph with 7 vertices is possible. The sum of the degrees of a regular graph's vertices must be even, but in this case, it's 21 (7*3), which is odd. Thus, it is not possible to have a group of 7 people where each is friends with exactly 3 others.
Step-by-step explanation:
The question is about determining whether it's possible to have a group of 7 people where each person is friends with exactly 3 other people in the group. This problem can be modeled using graph theory, a branch of mathematics, where each person is represented by a vertex (or node) and friendships between them are represented by edges (or lines) that connect these vertices.
In graph theory, this is known as finding a regular graph of degree 3 on 7 vertices. A graph is regular if all vertices have the same degree, which in this context means the same number of friends. For this question, we need to find a 3-regular graph (also called cubic graph) with 7 vertices. The challenge is to draw seven circles (representing the 7 people), where each circle has exactly 3 lines connecting it to other circles without any of the lines crossing.
To visualize this, one might try drawing diagrams or using graph theory software. However, by the definition of a regular graph, the sum of the degrees of all vertices must be an even number since each edge contributes to the degree of exactly two vertices. Here, 7 vertices each with a degree of 3 gives a sum of 21, which is not even. Therefore, it is impossible to have such a group where each person is friends with exactly 3 other people in the group.