Question

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

• A. 5
• B. 7
• C. 6
• D. 8

Hint:

In combinatorics, the difference between \( T_{n+1} \) and \( T_n \) often involves the difference between binomial coefficients.

Answer 1

The Correct Option is B

The number of triangles that can be formed with \( n \) vertices of a polygon is given by \( T_n = \binom{n}{3} \). We are given that \( T_{n+1} - T_n = 21 \), which gives: \[ \binom{n+1}{3} - \binom{n}{3} = 21 \] Simplifying this equation, we find that \( n = 7 \).