A convex polyhedron has $14$ vertices and $24$ edges. Each vertex is cut off by a plane that meets only the edges at that vertex, and no two of the cutting planes meet inside or on the polyhedron. How many faces does the resulting polyhedron have?

1. By Euler's formula the original polyhedron has $F=2-V+E=2-14+24=12$ faces.
2. Each cut removes one vertex and creates one new polygonal face; the $12$ original faces remain (each loses its corners).
3. The new polyhedron has $12+14=\boxed{26}$ faces. (Check: it has $V'=2E=48$ vertices and $E'=3E=72$ edges, and $2-48+72=26$.)