Find the value of P, Q, R and S.
`{:("Shapes","Sum of number of","Difference of number"),(,"faces and vertices","of edges and vertices"),("Hexagonal Prism"," "P," "Q),("Pentagonal Pyramid"," "R," "S):}`

Options

A. `{:(P,Q,R,S),(20,6,12,4):}`
B. `{:(P,Q,R,S),(20,12,6,6):}`
C. `{:(P,Q,R,S),(20,12,6,9):}`
D. `{:(P,Q,R,S),(8,12,6,12):}`

Answer

To solve the problem, we need to find the values of P, Q, R, and S for the given shapes: a Hexagonal Prism and a Pentagonal Pyramid.

### Step-by-Step Solution:

1. **Hexagonal Prism**:
- A hexagonal prism has two hexagonal bases. Each hexagon has 6 vertices.
- Therefore, the total number of vertices (V) in the hexagonal prism is:
\[ V = 6 \text{ (top hexagon)} + 6 \text{ (bottom hexagon)} = 12 \]

2. **Counting Edges**:
- The edges consist of the edges of the two hexagons and the edges connecting the corresponding vertices of the two hexagons.
- Each hexagon has 6 edges, and there are 6 vertical edges connecting the two hexagons.
- Therefore, the total number of edges (E) is:
\[ E = 6 \text{ (top hexagon)} + 6 \text{ (bottom hexagon)} + 6 \text{ (vertical edges)} = 18 \]

3. **Counting Faces**:
- The hexagonal prism has 2 hexagonal faces and 6 rectangular faces.
- Therefore, the total number of faces (F) is:
\[ F = 2 \text{ (hexagonal faces)} + 6 \text{ (rectangular faces)} = 8 \]

4. **Sum of Faces and Vertices**:
- Now, we calculate the sum of the number of faces and vertices:
\[ P = F + V = 8 + 12 = 20 \]

5. **Difference of Edges and Vertices**:
- Next, we find the difference between the number of edges and vertices:
\[ Q = E - V = 18 - 12 = 6 \]

6. **Pentagonal Pyramid**:
- A pentagonal pyramid has a pentagonal base with 5 vertices and 1 apex vertex.
- Therefore, the total number of vertices (R) is:
\[ R = 5 \text{ (base vertices)} + 1 \text{ (apex)} = 6 \]

7. **Counting Edges in the Pentagonal Pyramid**:
- The edges consist of the edges of the pentagon and the edges connecting the apex to each vertex of the pentagon.
- Therefore, the total number of edges (S) is:
\[ S = 5 \text{ (base edges)} + 5 \text{ (edges to apex)} = 10 \]

8. **Counting Faces of the Pentagonal Pyramid**:
- The pentagonal pyramid has 1 pentagonal face and 5 triangular faces.
- Therefore, the total number of faces (F) is:
\[ F = 1 \text{ (pentagonal face)} + 5 \text{ (triangular faces)} = 6 \]

9. **Sum of Faces and Vertices for Pentagonal Pyramid**:
- Now, we calculate the sum of the number of faces and vertices:
\[ R = F + V = 6 + 6 = 12 \]

10. **Difference of Edges and Vertices for Pentagonal Pyramid**:
- Finally, we find the difference between the number of edges and vertices:
\[ S = E - V = 10 - 6 = 4 \]

### Summary of Values:
- \( P = 20 \)
- \( Q = 6 \)
- \( R = 12 \)
- \( S = 4 \)