Animated Solution for Mathematics - Permutations and Combinations: Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is :

To form a triangle within a regular polygon with $n$ vertices, we must select any $3$ vertices from the $n$ available. Since the order of selection does not change the resulting triangle, we utilize the combination formula.

The total number of triangles $T_n$ is given by:

This expression serves as the bedrock of our logical framework.

We now consider a polygon with $n+1$ vertices. The number of triangles in this polygon is $T_{n+1}=\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)$. We are given the condition:

Rather than expanding these terms into complex factorials, we apply Pascal's Identity, which states:

By rearranging this identity, we obtain the relationship $\left(\genfrac{}{}{0ex}{}{n+1}{r}\right)-\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)$. Setting $r=3$, our equation simplifies elegantly to:

We are left with the simplified combination equation $\left(\genfrac{}{}{0ex}{}{n}{2}\right)=10$. Expanding this using the definition of combinations, we get:

Multiplying both sides by $2$, we arrive at the quadratic form:

We seek two consecutive integers whose product is $20$. Since $5\times 4=20$, we conclude that $n=5$.

The polygon in question is a pentagon. This result demonstrates that in JEE mathematics, identifying the correct combinatorial property often bypasses the need for tedious brute-force calculation.