A) \[3{{p}^{2}}(p-1)+1\]
B) \[3{{p}^{2}}(p-1)\]
C) \[{{p}^{2}}(4p-3)\]
D) \[{{p}^{2}}(2p-3)\]
Correct Answer: C
Solution :
| Maximum number of triangles \[{}^{3P}{{C}_{3}}-3({}^{P}{{C}_{3}})\] |
| \[=\frac{3p\,(3p-1)(3p-2)}{6}-\frac{3p\,(p-1)(p-2)}{6}\] |
| \[=\frac{p}{2}[(3p-1)(3p-2)-(p-1)(p-2)]\] |
| \[=\frac{p}{2}[9{{p}^{2}}-9p+2-{{p}^{2}}+3p-2]\] |
| \[=\frac{p}{2}[8{{p}^{2}}-6p]={{p}^{2}}(4p-3)\] |
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