SOLUTION: 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 th

Algebra ->  Formulas -> SOLUTION: 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 th      Log On


   

Question 1193015: 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)
Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
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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

         C%5Bn%5D%5E2 = %28n%2A%28n-1%29%29%2F2.


     Each pair of points determines a unique line.


     For example,  for 4 points  (n = 4),  there are  C%5B4%5D%5E2 = %284%2A%284-1%29%29%2F2 = %284%2A3%29%2F2 = 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

         C%5Bn%5D%5E3 = n%2A%28n-1%29%2A%28n-2%29%2F%281%2A2%2A3%29.


     Each triple of points determines a unique plane.


     For example,  for 4 points  (n = 4),  there are  C%5B4%5D%5E3 = %284%2A3%2A2%29%2F%281%2A2%2A3%29 = 4 planes.


     Compare it with 4 faces of a tetrahedron.

Solved and explained.