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1.Suppose you have n points, no three of which are collinear. How many lines contain two of these n points?
2.If no four of the n points are coplanar, how many planes contain three of the n points?
Hint: (for 1 and 2, generalize in a form of a formula)
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(1) The number of lines is the number of combinations n points taken 2 at a time
= .
Each pair of points determines a unique line.
For example, for 4 points (n = 4), there are = = = 2*3 = 6 lines.
Compare it with 4 sides + 2 diagonals of an arbitrary quadrilateral
(2) The number of planes is the number of combinations of n points taken 3 at a time
= .
Each triple of points determines a unique plane.
For example, for 4 points (n = 4), there are = = 4 planes.
Compare it with 4 faces of a tetrahedron.
Solved and explained.