Question

Can a polyhedron have $$10$$ faces, $$20$$ edges and $$15$$ vertices ?

Answer 1

$$\textbf{Step 1 - Check given condition is satisfied the Euler's formula or not}$$

$$\text{We know Euler;s formula for a polyhedron}$$

$$\textbf{ [Vertices(V)-Edges(E)+Faces(F)=2]}$$

$$\text{Given, F=10, V=15, E=20}$$

$$\Rightarrow$$ $$ 15-20+10=5$$

$$\text{Which does not satisfy the Euler's formula.}$$

$$\textbf{Hence, a polyhedron cannot have 10 faces,20 edges and 15 vertices.}$$