A) \[5\]
B) \[7\]
C) \[6\]
D) \[4\]
Correct Answer: B
Solution :
Key Idea: For making a triangle, we needs a three vertices of a triangle. \[{{T}_{n+1}}-{{T}_{n}}=21\] \[\Rightarrow \] \[^{n+1}{{C}_{3}}{{-}^{n}}{{C}_{3}}=21\,\,(\because \,\,{{T}_{n}}{{=}^{n}}{{C}_{3}})\] \[\Rightarrow \] \[^{n}{{C}_{2}}{{+}^{n}}{{C}_{3}}{{-}^{n}}{{C}_{3}}=21\] \[\Rightarrow \] \[^{n}{{C}_{2}}=21\] \[\Rightarrow \] \[\frac{n(n-1)}{2}=21\] \[\Rightarrow \] \[{{n}^{2}}-n-42=0\] \[\therefore \] \[n=7\]
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