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There are 4 points in space, not all lying in the same plane. How many distinct parallelepipeds can be constructed, if these 4 points are on its 8 vertices? Please I need a little help, how to start.

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    I think for one parallelepiped we need 8 points but here only 4 points are given – 2017-01-04

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  1. First of all - choose 4 vertices in ${8\choose 4}$ ways

  2. Exclude situations, when all selected vertices are on the same side ($6$ ways) or on two opposite (not lying on the same side) parallel edges ($6$ ways)

  3. The final answer is then $$n={8\choose 4} - 12 = 58$$