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Checking using Euler's Formula: F + V = E +2; Given that F = 10, V = 15 and E = 20 so, Putting in equation: 10 + 15 = 35  and 20 +2 = 22 Hence, given polyhedron having 10 faces, 20 edges and 15 vertices is not possible.
We know the Euler Formula: F + V = E + 2: hence we can easily find the unknown let say 'F, V, and E' So, For Ist case:  F + 6  = 12 + 2  or  F = 8, hence the number of faces in Ist case is 8. For IInd Case; 5 + V = 9 + 2   or  V = 6, hence the number of vertices in IInd case is 6. For IIIrd case; 20 + 12 = E + 2     or   E = 30, hence the number of Edges in the IIIrd case is...
(ii) Euler formula F+V = E + 2. So, Number of faces (F) = 9, Number of vertices (V)  = 9 and number of edges (E)  = 16. Hence putting in the formula; we get, 9 + 9 = 16 + 2 18 = 18 hence, it satisfies the Euler formula.
(i) Euler formula F+V = E + 2. So, Number of faces (F) = 7, Number of vertices (V)  = 10 and number of edges (E)  = 15. Hence putting in the formula; we get, 7 + 10 = 15 + 2 17 = 17 hence, it satisfies the Euler formula.  
No, not always, because it can be a cuboid also when extended.
(i) A prism is a polyhedron whose base and top are congruent polygons and whose other faces, i.e., lateral faces are parallelograms in shape. Also, the cylinder has base on top and bottom and faces are parallel.
(iii) Yes, a polyhedron can have faces of a square and four triangles which makes a pyramid on square base.
(ii) Yes, a polyhedron having 4 triangles is pyramid on triangular base.
(i) No, a polyhedron cannot have 3 triangles for its faces. 
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