Let T_n denote the number of triangles which | vedclass

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Question

(B) The number of triangles that can be formed using the vertices of a regular polygon of $n$ sides is given by the combination formula $T_n = ^nC_3$.

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(B) The number of triangles that can be formed using the vertices of a regular polygon of $n$ sides is given by the combination formula $T_n = ^nC_3$.Given the equation $T_{n + 1} - T_n = 21$,we substitute the formula:$^{n + 1}C_3 - ^nC_3 = 21$Using the Pascal's identity property of combinations,$^nC_r + ^nC_{r-1} = ^{n+1}C_r$,we know that $^{n+1}C_3 = ^nC_3 + ^nC_2$.Substituting this into the equation:$(^nC_3 + ^nC_2) - ^nC_3 = 21$$^nC_2 = 21$Expanding the combination formula:$\frac{n(n - 1)}{2} = 21$$n(n - 1) = 42$$n^2 - n - 42 = 0$$(n - 7)(n + 6) = 0$Since $n$ must be a positive integer,$n = 7$.

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

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