Permutations and Combinations
Geometrical Applications
IIT-JEE 2001 Screening
Grade 11
Question:
Let $T_n$ denote the number of triangles formed using vertices of a regular polygon of $n$ sides. If $T_{n+1} - T_n = 21$, then $n$ equals:
(1) $3$
(2) $5$
(3) $7$
(4) $9$
Step-by-Step Solution
Key Concept: $T_n = ^nC_3$. Then $T_{n+1} - T_n = ^{n+1}C_3 - ^nC_3 = ^nC_2$ (Pascal). So $^nC_2 = 21 \Rightarrow n(n - 1) = 42 \Rightarrow n = 7$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)