Let T_n be the number of all possible triangl | vedclass

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(B) The number of triangles formed by joining the vertices of an $n$-sided polygon is given by the combination formula $T_n = ^nC_3$.Given the conditi

Vedclass · Accepted Answer

$5$

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(B) The number of triangles formed by joining the vertices of an $n$-sided polygon is given by the combination formula $T_n = ^nC_3$.Given the condition $T_{n+1} - T_n = 10$,we substitute the formula:$^{n+1}C_3 - ^nC_3 = 10$Using the property of combinations,we know that $^nC_r + ^nC_{r-1} = ^{n+1}C_r$,which implies $^{n+1}C_3 - ^nC_3 = ^nC_2$.Therefore,$^nC_2 = 10$.Expanding the combination formula:$\frac{n(n-1)}{2} = 10$$n(n-1) = 20$$n^2 - n - 20 = 0$$(n-5)(n+4) = 0$Since $n$ represents the number of sides of a polygon,$n$ must be a positive integer. Thus,$n = 5$.

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:

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