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Question

As shown in the diagram, points p subscript 1 end subscript comma p subscript 2 end subscript comma p subscript 3 end subscript horizontal ellipsis horizontal ellipsis horizontal ellipsis. p subscript 10 end subscript are either the Then the number of groups of four points

open parentheses p subscript 1 end subscript comma p subscript i end subscript comma p subscript j end subscript comma p subscript k end subscript close parentheses left parenthesis 1 less than i less than j less than k £ 10 right parenthesis on the same plane is.

  1. 30    
  2. 33    
  3. 36    
  4. 39    

The correct answer is: 33


    On each laternal face of the tetrahedron other than p subscript 1 end subscript there are 5 pointsTake any 3 out of these 5 and add p subscript 1 end subscript to it(e.g open parentheses p subscript 1 end subscript p subscript 3 end subscript p subscript 4 end subscript p subscript 6 end subscript close parentheses So, there are in all 35 subscript C subscript 1 end subscript end subscript groups of laternal facesApart from these, there are 3 points on each edge containing p subscript 1 end subscript When we add a mid point taking from the edge on the base which is not on the same plane with the edge abovewe obtain another required group (.... e.gp p subscript 1 end subscript p subscript 2 end subscript p subscript 6 end subscript p subscript 10 end subscript ) There 3 groups like this.
    So, total required groups equals 3 to the power of ´ end exponent 5 subscript C subscript j end subscript end subscript plus 3 equals 33

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