GATE | GATE-CS-2014-(Set-1) | Question 65

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? _____________
(A) 6
(B) 8
(C) 4
(D) 10


Answer: (A)

Explanation: The Tetrahedron has 4 triangular surfaces with 4 vertices/corners ( say A,B,C and D) as can be seen here. http://www.sjsu.edu/faculty/wa…

Now if you take a point inside a tetrahedron (suppose O) and connect it with any two of its corners which are nothing but vertices( suppose A and B), you will get 1 internal plane as OAB.

So we can see from here that, no of new internal planes = no of different pair of corners or vertices

Similarly you can take any other 2 corners like (A,C) or (A,D) or (B,C) or (B,D) or (C,D),
hence total possible pair of corners are 6. Therefore 6 new internal planes possible.

We could also calculate the possible corners by using combinations formula,
which is nCr, i.e. no of ways to select a combination of r things from a given set of n things.
here n = 4 ( as total 4 vertices, A,B,C and D)
and r =2 ( as we need two corners at a time)
Thus, 4C2 = 6.

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