Short Answer 
Euler’s characteristic formula (V – E + F = 2) helps analyze the structure of a football, identifying it as a truncated icosahedron. By calculating the total faces (F = 32) and edges (E = 90), we find the number of vertices (V) to be 60, confirming that a typical football has 60 vertices.
Step 1: Understand Euler’s Characteristic Formula
To analyze the structure of a football, also known as a truncated icosahedron, we utilize Euler’s characteristic formula, expressed as V – E + F = 2. Here, V represents the number of vertices, E represents edges, and F represents faces. This formula is essential for calculating the geometric properties of polyhedra, including the configuration of a football made up of pentagons and hexagons.
Step 2: Count Faces and Calculate Edges
For our football, we determine the total faces (F) as the sum of the two types of polygons: 12 pentagons and 20 hexagons, giving us F = 12 + 20 = 32. Next, we calculate the total edges: each pentagon contributes 5 sides (5 x 12), and each hexagon contributes 6 sides (6 x 20), leading to a total of 60 + 120 = 180 sides. Since each edge is shared between two faces, the unique edges (E) result in E = 180 / 2 = 90 edges.
Step 3: Solve for Vertices
Now, we substitute our calculated values into Euler’s formula to find the vertices (V). Using V = E – F + 2, we plug in E = 90 and F = 32 to get V = 90 – 32 + 2. Performing the calculation yields V = 60. Thus, the final conclusion is that a typical football has 60 vertices, confirming the correct answer is option (c) 60.