$T_m$ 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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Aicharya · Accepted Answer

Given, $T_m=$ Number of triangles formed with the vertices of a polygon of $m$ sides. Also, $T_{m+1}-T_m=15$ As we know, ${ }^n C_r+{ }^n C_{r+1}={ }^{n+1} C_{r+1}$ $	herefore \quad{ }^m C_2+{ }^m C_3={ }^{m+1} C_{2+1}$ ${ }^m C_2+{ }^m C_3=15+{ }^m C_3 \quad$ [from eq. (i) $]$ $\begin{aligned} & 	herefore \quad{ }^m C_2=15 \ & \frac{m !}{2 !(m-2) !}=15 \ & m(m-1)=30 \ & m^2-m-30=0 \ & (m-6)(m+5)=0 \ & m=6-5\end{aligned}$ $	herefore m=6 \quad[\because m 
eq-5]$

$T_m$ denotes the number of triangles that can be formed with the vertices of a regular polygon of $m$ sides…

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