Problem:
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
Answer Choices:
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Solution:
A plane that intersects at least three vertices of a cube either cuts into the cube or is coplanar with a cube face. Therefore the three randomly chosen vertices result in a plane that does not contain points inside the cube if and only if the three vertices come from the same face of the cube. There are cube faces, so the number of ways to choose three vertices on the same cube face is . The total number of ways to choose the distinct vertices without restriction is . Hence the probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions