NIMCET 2013 — Mathematics PYQ

Step 1: Understand the formula for $T_n$

A triangle is formed by selecting any $3$ vertices from the $n$ available vertices of the polygon. The number of ways to choose $3$ points out of $n$ is given by the combination formula ${}^n{C}_3$.

Step 2: Set up the given equation

We are given that $T_{n+1}-T_n=21$. Substituting the combination formula:

Step 3: Simplify using Pascal's Identity

Recall the property of combinations: $\left(\genfrac{}{}{0ex}{}{n}{r}\right)+\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)=\left(\genfrac{}{}{0ex}{}{n+1}{r}\right)$.

This can be rearranged as:

So, the equation becomes:

Step 4: Solve for $n$

Expand the combination:

Step 5: Factorize the quadratic equation

Looking for two numbers that multiply to $-42$ and add to $-1$:

Since $n$ represents the number of sides of a polygon, it must be a positive integer.

Final Answer:

The value of $n$ is 7.