Every vertex in a graph G has either degree 1 or degree 3. Knowing that the graph has 10 edges and exactly 2 nodes of degree 3 show that it cannot have 5 connected components.

To analyze the problem, let’s break down the given information:

1. **Degrees of vertices**: In graph \( G \), every vertex has either degree 1 or degree 3.
2. **Total edges**: The graph has 10 edges.
3. **Vertices of degree 3**: There are exactly 2 vertices of degree 3.

Let’s denote:
- \( n_1 \): the number of vertices of degree 1.
- \( n_3 = 2 \): the number of vertices of degree 3.

### Step 1: Calculate the total degree
The sum of the degrees of all vertices in a graph is equal to twice the number of edges. Therefore, we can write:

\[
\text{Total degree} = 2 \times \text{Number of edges} = 2 \times 10 = 20.
\]

The total degree can also be expressed in terms of the degrees of the vertices:

\[
\text{Total degree} = 1 \times n_1 + 3 \times n_3 = n_1 + 3 \times 2 = n_1 + 6.
\]

Setting these two expressions for total degree equal gives:

\[
n_1 + 6 = 20.
\]

Solving for \( n_1 \):

\[
n_1 = 20 - 6 = 14.
\]

### Step 2: Count the total number of vertices
The total number of vertices \( n \) in the graph is the sum of the vertices of degree 1 and degree 3:

\[
n = n_1 + n_3 = 14 + 2 = 16.
\]

### Step 3: Components and edges
Now, consider the connectivity of the graph. A connected component is formed by vertices connected through edges. 

For \( k \) components, the number of edges \( E \) can be calculated as follows. The minimum number of edges required to connect \( n \) vertices in \( k \) components is:

\[
E_{\text{min}} = n - k.
\]

In our case, since \( E = 10 \) and \( n = 16 \):

\[
E_{\text{min}} = 16 - k.
\]

Setting this equal to the given number of edges:

\[
16 - k \leq 10 \implies k \geq 6.
\]

### Conclusion
Since \( k \) must be at least 6, having 5 connected components (i.e., \( k = 5 \)) contradicts this condition. Therefore, the graph cannot have 5 connected components.

### Summary
We have shown that a graph \( G \) with 10 edges, where every vertex has either degree 1 or degree 3, and exactly 2 vertices of degree 3, cannot have 5 connected components. 

Would you like further details or have any questions? Here are some related questions for consideration:

1. How would the scenario change if the number of edges were increased?
2. What would happen if there were 3 vertices of degree 3 instead of 2?
3. Can you describe the structure of a possible graph with the given conditions?
4. What are the implications of having vertices with higher degrees?
5. How does the connectivity of a graph affect its components?

**Tip**: Always verify the relationship between edges and vertices when dealing with graph components; it can reveal constraints on the structure of the graph.

Explore the properties of a graph where each vertex has a degree of 1 or 3, with 10 edges and 2 vertices of degree 3. This analysis demonstrates why the graph cannot have 5 connected components. Ideal for Grades 10-12 students studying graph theory.

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