Combinatorics of a pentagonal prism\nA pentagonal prism has 15 edges. How many vertices does it have?

Introduction / Context:
For a right or oblique prism based on an n-gon, counts are standard: vertices = 2n, edges = 3n, faces = n + 2. Recognizing this pattern avoids lengthy enumeration.

Given Data / Assumptions:

• Pentagonal prism ⇒ n = 5
• Edges = 3n = 15 (consistent)

Concept / Approach:
Use the prism formula for vertices: V = 2n. Substitute n = 5 to get the vertex count directly. These relations can also be derived from Euler’s formula V − E + F = 2 with known face counts.

Step-by-Step Solution:

Verification / Alternative check:
Faces F = n + 2 = 7. Check Euler: V − E + F = 10 − 15 + 7 = 2 (valid), confirming counts.

Why Other Options Are Wrong:
12, 15, and 20 correspond to different polyhedra or misapplied formulas; only 10 satisfies the prism identities.

Common Pitfalls:
Confusing prisms with pyramids (where counts differ) or forgetting that there are two congruent n-gon bases leads to errors.

Final Answer:
10