T_m denotes the number of triangles that can | vedclass

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

Vedclass · Accepted Answer

$6$

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(B) The number of triangles that can be formed using the vertices of a regular polygon of $m$ sides is given by $T_m = {}^{m}C_3$.Given the condition $T_{m+1} - T_m = 15$,we substitute the formula:${}^{m+1}C_3 - {}^{m}C_3 = 15$Using the identity ${}^{n}C_r = {}^{n-1}C_r + {}^{n-1}C_{r-1}$,we know that ${}^{m+1}C_3 - {}^{m}C_3 = {}^{m}C_2$.Therefore,${}^{m}C_2 = 15$.Expanding the combination formula: $\frac{m(m-1)}{2} = 15$.$m^2 - m = 30$.$m^2 - m - 30 = 0$.$(m-6)(m+5) = 0$.Since $m$ must be a positive integer,$m = 6$.

$T_m$ denotes the number of triangles that can be formed with the vertices of a regular polygon of $m$ sides. If $T_{m+1}-T_m=15$,then $m$ is equal to

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