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Let $T_{n}$ denotes the number of triangles which can be formed using the vertices of a polygon of n sides. If $T_{n +1} {- T}_{n} = 21,$ then n = __

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Detailed Solution

$T_{n+1} {- T}_{n} = 21$

$\Rightarrow {(n + 1)C}_{3} {- nC}_{3} = 21$

$\Rightarrow \frac{(n + 1) (n) (n - 1)}{6} - \frac{n (n - 1) (n - 2)}{6} = 21$

$\Rightarrow n (n - 1) = 42 \Rightarrow n = 7$

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