Question

\( T_m \) 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 \( m = \)

• A. \( 3 \)
• B. \( 6 \)
• C. \( 9 \)
• D. \( 12 \)

Hint:

Key identity: \( \binom{n+1}{r} - \binom{n}{r} = \binom{n}{r-1} \). For \( r=3 \), \( \binom{m+1}{3} - \binom{m}{3} = \binom{m}{2} \).

Answer 1

The Correct Option is B

Concept: Number of triangles from vertices of an \( m \)-sided polygon = \( \binom{m}{3} \), since any 3 non-collinear vertices form a triangle.

Step 1: Write the given condition. \[ T_m = \binom{m}{3}, \quad T_{m+1} = \binom{m+1}{3}. \] Given: \[ \binom{m+1}{3} - \binom{m}{3} = 15. \]

Step 2: Use the identity \( \binom{m+1}{3} - \binom{m}{3} = \binom{m}{2} \). \[ \binom{m}{2} = 15. \]

Step 3: Solve for \( m \). \[ \frac{m(m-1)}{2} = 15 \quad \Rightarrow \quad m(m-1) = 30. \] \[ m^2 - m - 30 = 0 \quad \Rightarrow \quad (m-6)(m+5) = 0. \] \[ m = 6 \quad (\text{since } m > 0). \]