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Question
Can a polyhedron have 10 faces, 20 edges and 15 vertices?
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Solution
\[\text { No, because every polyhedron satisfies Euler's formula, given below: }\]
\[\text { F+V=E+2 }\]
\[\text { Here, number of faces F = 10 }\]
\[\text { Number of edges E = 20 }\]
\[\text { Number of vertices V = 15 }\]
\[\text { So, by Euler's formula: }\]
\[\text { LHS: 10+15 = 25 }\]
\[\text { RHS: 20 + 2 = 22 }, \]
\[\text { which is not true because 25 }\neq22\]
\[\text { Hence, Eulers formula is not satisfied and no polyhedron may be formed .}\]
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