Color each vertex of the pyramid S-ABCD so that no two vertices on the same edge have the same color. If there are 5 colors available, how many different ways are there to color the vertices?

To solve this problem, we need to find how many different ways we can color the vertices of the pyramid \( S-ABCD \) with the restriction that no two adjacent vertices (those connected by an edge) share the same color. There are 5 available colors, and the structure of the pyramid forms specific restrictions based on the edges connecting the vertices.

### Step 1: Understanding the Pyramid Structure
The pyramid has 5 vertices:
- The apex \( S \),
- The base vertices \( A, B, C, D \), which form a quadrilateral (usually a square or some polygon).

Edges of the pyramid:
- \( S \) is connected to each of the base vertices \( A, B, C, D \), giving us 4 edges: \( SA, SB, SC, SD \).
- The base is a quadrilateral, so \( A, B, C, D \) are connected by 4 edges: \( AB, BC, CD, DA \).

Thus, there are 8 edges in total:
- 4 from the apex to the base vertices,
- 4 forming the quadrilateral at the base.

### Step 2: Analyzing the Coloring Problem
We need to color the 5 vertices \( S, A, B, C, D \) such that no two vertices connected by an edge share the same color.

### Step 3: Color Assignment
- **Vertex \( S \)**: Since no restrictions exist yet, we can choose any of the 5 available colors for vertex \( S \).
- **Vertex \( A \)**: Vertex \( A \) is connected to vertex \( S \), so we must assign it a color different from \( S \). Thus, we have 4 remaining color choices for \( A \).
- **Vertex \( B \)**: Vertex \( B \) is connected to both \( A \) and \( S \), so we must choose a color for \( B \) that is different from both. Therefore, we have 3 remaining color choices for \( B \).
- **Vertex \( C \)**: Vertex \( C \) is connected to \( B \), \( A \), and \( S \), so we need to choose a color for \( C \) that differs from all three. This leaves 2 remaining color choices for \( C \).
- **Vertex \( D \)**: Vertex \( D \) is connected to \( C \), \( B \), \( A \), and \( S \), so we must assign it the remaining available color. Only 1 color remains for \( D \).

### Step 4: Total Number of Ways to Color the Vertices
The total number of ways to assign colors is the product of the choices for each vertex:
\[
5 \times 4 \times 3 \times 2 \times 1 = 120.
\]

Thus, the total number of different ways to color the vertices is \( \boxed{120} \).

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Do you want further details or have any questions?

Here are five questions to expand on this topic:
1. How would the number of ways change if only 4 colors were available instead of 5?
2. How many ways can you color the vertices if two adjacent vertices are allowed to share the same color?
3. How does the shape of the base (square, triangle, etc.) affect the number of colorings?
4. What if the pyramid had an additional vertex and edge structure?
5. How would you calculate the number of ways to color a more complex 3D shape?

**Tip**: When solving coloring problems, always check for symmetry and adjacency relations carefully to avoid overcounting or violating the problem's constraints.

Learn how to solve the vertex coloring problem for the pyramid S-ABCD where no two vertices connected by an edge share the same color. Using combinatorics and graph theory, discover how 5 colors lead to 120 different ways of coloring the vertices. Suitable for students in Grades 9-12.

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