How many triangles (T_n) can be formed by joi | vedclass

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(B) The number of triangles that can be formed by joining the vertices of a regular polygon with $n$ sides is given by the combination formula $T_n =

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$5$

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(B) The number of triangles that can be formed by joining the vertices of a regular polygon with $n$ sides is given by the combination formula $T_n = \binom{n}{3} = \frac{n(n-1)(n-2)}{6}$.Given the condition $T_{(n+1)} - T_n = 10$,we substitute the formula:$\binom{n+1}{3} - \binom{n}{3} = 10$.Using the identity $\binom{n+1}{k} - \binom{n}{k} = \binom{n}{k-1}$,we get:$\binom{n}{2} = 10$.$\frac{n(n-1)}{2} = 10$.$n(n-1) = 20$.$n^2 - n - 20 = 0$.$(n-5)(n+4) = 0$.Since $n$ must be a positive integer,$n = 5$.

How many triangles $(T_n)$ can be formed by joining the vertices of a regular polygon with $n$ sides? If $T_{(n+1)} - T_n = 10$,what is the value of $n$?

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