A) \[{{p}^{3}}+3{{p}^{2}}\]
B) \[\frac{1}{2}({{p}^{3}}+p)\]
C) \[\frac{{{p}^{2}}}{2}(5p-3)\]
D) \[{{p}^{2}}(4p-3)\]
Correct Answer: D
Solution :
Total number of points in a plane is 3P. \[\therefore \]Maximum number of triangles \[{{=}^{3P}}{{C}_{3}}-{{3.}^{P}}{{C}_{3}}\] (here, we subtract those triangles which points are in a line) \[=\frac{3P(3P-1)(3P-2)}{3\times 2}-\frac{3\times P(P-1)(P-2)}{3\times 2}\] \[=\frac{P}{2}[9{{P}^{2}}-9P+2-({{P}^{2}}-3P+2)]\] \[={{P}^{2}}(4P-3)\]
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