When a point inside of a tetrahedron (a solid with four triangular surfaces) is connected by straight lines to its corners, how many (new) internal planes are created with these lines?

GATE 2014 · General Aptitude · Geometry · medium

Answer: 6 new internal planes are created.

1. Count edges of the tetrahedron: Edges = C(4,2) = 4!/(2! x 2!) = 6. The edges are AB, AC, AD, BC, BD, CD.
2. Each edge with point P defines one new internal plane: Point P inside the tetrahedron, together with each edge (two vertices), defines a unique plane: PAB, PAC, PAD, PBC, PBD, PCD. That is 6 planes total.