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 \[{{=}^{3p}}{{C}_{3}}-3{{\cdot }^{p}}{{C}_{3}}\] [Here, we subtract those triangles which points are in a line] \[=\frac{(3p)!}{(3p-3)!3!}-3.\frac{p!}{(p-3)!3!}\] \[=\frac{3p(3p-1)(3p-2)}{3\times 2}-\frac{3\times p(p-1)(p-2)}{3\times 2}\] \[=\frac{p}{2}[9{{p}^{2}}-9p+2-({{p}^{2}}-3p+2)]\] \[={{p}^{2}}[4p-3]\]
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