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  • Let denote the number of triangles which can be formed by using the vertices of a regular

Let \mathrm{T_{n}} denote the number of triangles which can be formed by using the vertices of a regular polygon of \mathrm{n} sides. If \mathrm{T_{n+1}-T_{n}=21}, then \mathrm{n} equals

Option: 1

5


Option: 2

7


Option: 3

6


Option: 4

4


Answers (1)

best_answer

The number of triangles that can be formed by using the vertices of a regular polygon is \mathrm{{ }^{n} C_{3}}. That is , \mathrm{T_{n}={ }^{n} C_{3}}

Now, \mathrm{\quad T_{n+1}-T_{n}=21}

\mathrm{\Rightarrow \quad{ }^{n+1} C_{3}-{ }^{n} C_{3}=21}
\mathrm{\Rightarrow{ }^{n} C_{2}+{ }^{n} C_{3}-{ }^{n} C_{3}=21}      \mathrm{\left[\because{ }^{n+1} C_{r}={ }^{n} C_{r-1}+{ }^{n} C_{r}\right]}

\mathrm{\Rightarrow \quad \frac{1}{2} n(n-1)=21\quad n= -6\, or\, 7}.    

As \mathrm{n} is a positive integer, \mathrm{n= 7}.

Posted by

Gautam harsolia

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