A solid three dimensional body has 12 vertices and 30 edges. Using Euler's formula for polyhedra, how many faces does this solid have in total?

Introduction / Context:
This question tests your understanding of Euler's formula for convex polyhedra, a fundamental concept in solid geometry. The relationship links the number of vertices, edges and faces of any convex polyhedron. Many competitive exams ask direct questions of this form because once you know the formula, the calculation is very quick and reliable.

Given Data / Assumptions:
- Number of vertices V = 12.
- Number of edges E = 30.
- The solid is assumed to be a convex polyhedron so that Euler's formula applies directly.
- We need to find the total number of faces F of this solid.

Concept / Approach:
Euler's famous formula for a convex polyhedron states that:
V - E + F = 2
where V is the number of vertices, E is the number of edges and F is the number of faces. This equation is always satisfied for simple convex solids such as prisms, pyramids and many more complex shapes as long as they remain topologically equivalent to a sphere. We will substitute the given values of V and E into this formula and solve for F.

Step-by-Step Solution:
Step 1: Write down Euler's formula: V - E + F = 2. Step 2: Substitute V = 12 and E = 30 into the formula. Step 3: We get 12 - 30 + F = 2. Step 4: Simplify 12 - 30 = -18, so the equation becomes -18 + F = 2. Step 5: Add 18 to both sides to obtain F = 2 + 18 = 20.

Verification / Alternative check:
You can rearrange Euler's formula explicitly as F = 2 - V + E. Substituting again gives F = 2 - 12 + 30 = 20. This quick check confirms our earlier calculation. Whenever you get an answer, it is good practice to verify it using a slightly rearranged version of the same formula to avoid simple arithmetic mistakes.

Why Other Options Are Wrong:
22, 24 and 26 do not satisfy Euler's relation when V = 12 and E = 30. For example, if F were 22 then V - E + F would be 12 - 30 + 22 = 4, which does not equal 2. Similar checks show that 24 and 26 also violate Euler's formula, so they cannot be correct.
18 would give 12 - 30 + 18 = 0, again not equal to 2, so it is also incorrect.

Common Pitfalls:
A frequent error is to mix up the signs when substituting into V - E + F = 2 or to use a wrong version of the formula. Some students also confuse polyhedra with planar graphs and try to apply graph theory ideas incorrectly. Always remember that for standard convex 3D solids, Euler's formula V - E + F = 2 is directly applicable and extremely reliable.

Final Answer:
The solid has 20 faces in total.