${\mathrm{T}}_{\mathrm{m}}$ denotes the number of Triangles that can be formed with the vertices of a regular polygon of m sides. If ${\mathrm{T}}_{\mathrm{m}+1} – {\mathrm{T}}_{\mathrm{m}} = 15$, then m = 
 

${\mathrm{T}}_{\mathrm{m}}=$ No of triangles that can be formed using the vertices of a regular polygon of m sides.

$\Rightarrow {\mathrm{T}}_{\mathrm{m}}={\mathrm{m}}_{{\mathrm{C}}_3}$

${\mathrm{T}}_{\mathrm{m}+1}-{\mathrm{T}}_{\mathrm{m}}=15$

${}^{(\mathrm{m}+1)}{\mathrm{C}}_3-{}^{\mathrm{m}}{\mathrm{C}}_3=15 \Rightarrow {}^{\mathrm{m}}{\mathrm{C}}_2=15$

$\frac{\mathrm{m}(\mathrm{m}-1)}{2}=15 \Rightarrow \mathrm{m}(\mathrm{m}-1)=30=6\times 5$

$\boxed{ \therefore \mathrm{m}=6}$