How many valid selections of 3 edges can be painted red on a cube, given the rule that a maximum of 2 edges can meet at a vertex?

To determine the number of valid selections of 3 edges that can be painted red on a cube, given that a maximum of 2 edges can meet at a vertex, we need to analyze the structure of a cube and the restrictions imposed by the problem.

A cube has 12 edges, and for any selection of edges, we need to ensure that no more than 2 edges can meet at any single vertex. In a cube, each vertex is connected to 3 edges. Therefore, if we select 3 edges, they cannot all be connected to the same vertex.

First, consider how to select 3 edges while respecting the limitation on vertices. One effective way to visualize edge selection is to examine combinations of edges that do not share vertices.

One valid approach is to use combinations of edges that are not adjacent. For instance, we can choose edges in different face sets or across the cube in a way that ensures compliance with the vertex meeting rule.

In a detailed combinatorial breakdown, we can note the following:

1. Selecting edges from non-adjacent faces can lead to valid combinations, as choosing edges purely from one face does not meet the requirement.

2. Considering symmetrical properties of the cube, we can multiply the valid configurations found