If T_n denotes the number of triangles which | vedclass

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(B) The number of triangles that can be formed using the vertices of a regular polygon of $n$ sides is given by $T_n = {}^{n}C_3$. Given the condition

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$7$

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(B) The number of triangles that can be formed using the vertices of a regular polygon of $n$ sides is given by $T_n = {}^{n}C_3$. Given the condition $T_{n+1} - T_n = 21$,we substitute the formula: ${}^{n+1}C_3 - {}^{n}C_3 = 21$. Using the identity ${}^{n}C_r + {}^{n}C_{r-1} = {}^{n+1}C_r$,we know that ${}^{n+1}C_3 - {}^{n}C_3 = {}^{n}C_2$. Therefore,${}^{n}C_2 = 21$. Expanding the combination formula: $\frac{n(n-1)}{2} = 21$. $n(n-1) = 42$. $n^2 - n - 42 = 0$. $(n-7)(n+6) = 0$. Since $n$ must be a positive integer,$n = 7$. Thus,option $(B)$ is correct.

If $T_n$ denotes the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides and $T_{n+1}-T_{n}=21$,then $n=$

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