Could you count lines?
Geometry
Level
3
Consider the 12 face diagonals of a rectangular block. How many
pairs of them are skew lines?
30
24
none of these
48
60
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1 solution
For any given face diagonal, there is exactly one face diagonal on each of the remaining faces to which the given face diagonal is skew. Therefore, for a given face diagonal D, there are 5 pairs of face diagonals that include D. Because there are 12 face diagonals, there will be ( 1 2 ) ( 5 ) = 6 0 pairs of skew lines. However, counting in this fashion double counts each pair, so we divide this result by 2 to conclude that we have 30 such pairs.