Let \(T_n\) be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If \(T_{n+1}\) – \(T_n\) = 10, then the value of n is

Solution :

Given, \(T_n\) = \(^nC_3\)

\(T_{n+1}\) = \(^{n+1}C_3\)

\(\therefore\) \(T_{n+1}\) – \(T_n\) = \(^{n+1}C_3\)  – \(^{n}C_3\)  = 10  [given]

\(\therefore\) \(^nC_2\) + \(^nC_3\) – \(^nC_3\) = 10

\(\implies\) \(^nC_2\) = 10

\(\therefore\) n = 5

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