Problem 164 - A Complete Resource Book in Mathematics for JEE Main

Question

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 (A) 5 (B) 10 (C) 8 (D) 7

Step-by-Step Solution

Answer

The value of \( n \) is 5.

Step 1: Understand the Problem

We need to find the value of \( n \) such that the difference between the number of triangles formed by joining vertices of an \( (n+1)\)-sided polygon, \( T_{n+1} \), and \( n \)-sided polygon, \( T_n \), is 10.

Step 2: Formula for Counting Triangles

The number of triangles that can be formed from an \( n \)-sided polygon by joining its vertices is given by the combination formula \( T_n = \binom{n}{3} \). This is because we need to pick 3 vertices out of \( n \) vertices to form a triangle.

Step 3: Set Up the Equation

According to the problem, \( T_{n+1} - T_n = 10 \). Using the triangular combination formula, we have \( \binom{n+1}{3} - \binom{n}{3} = 10 \).

Step 4: Expand the Combination Formula

Let's expand the combination terms: \( \binom{n+1}{3} = \frac{(n+1)n(n-1)}{6} \) and \( \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \).

Step 5: Simplify the Difference

The equation becomes:\[ \frac{(n+1)n(n-1)}{6} - \frac{n(n-1)(n-2)}{6} = 10 \]We simplify the left side by combining these fractions:\[ \frac{n(n-1)((n+1) - (n-2))}{6} = 10 \]This simplifies to \( \frac{n(n-1)(3)}{6} = 10 \), or \( \frac{n(n-1)}{2} = 10 \).

Step 6: Solve for n

The equation \( \frac{n(n-1)}{2} = 10 \) can be rewritten as \( n(n-1) = 20 \). Solving the quadratic equation \( n^2 - n - 20 = 0 \) using factoring or the quadratic formula, we find that \( n = 5 \) or \( n = -4 \). Since \( n \) must be a positive integer, we conclude that \( n = 5 \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, grouping, and arranging items. Think of it as the art and science of counting. When working with combinatorics, you often explore how items can be arranged into various configurations. This involves selecting a subset of items from a larger collection, like choosing team members from a group.

In our exercise, we are dealing with triangles formed from the vertices of a polygon. The real task is counting the number of triangles without manually drawing each one. Using combinatorics, we can rely on mathematical formulas to determine the number of possible triangles quickly. This is essential for handling large data sets where visualization would be cumbersome.

Regular Polygons

A regular polygon is a geometric figure with all sides and angles equal. Common examples include equilateral triangles and squares. When you see the term "regular polygon," remember uniformity and symmetry. Each vertex connects to others, creating opportunities for forming geometric shapes within the polygon.

In our problem scenario, we use regular polygons to count triangles. When vertices are plugged into combinatorial formulas, the regular polygon's properties make calculations straightforward—every combination of vertices is valid since all are equally spaced and connected.

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics, written as \( \binom{n}{k} \). It represents the number of ways to choose \( k \) items from a set of \( n \) items without regard to order. The formula is crucial:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

In this context, the binomial coefficient determines how many triangles we can form. For any \( n \)-sided polygon, we apply \( \binom{n}{3} \) to find all possible triangles because we select 3 vertices per triangle. The mathematical expression lets us count without drawing, significantly simplifying the work.

Polygon Geometry

Polygon geometry involves studying shapes formed by straight lines. Each polygon is defined by its vertices and edges. Geometry of polygons touches on various topics like symmetry, angles, and the shapes formed within.

In exercises like ours, polygon geometry becomes tangible when looking at triangles inside polygons. By examining how vertices interact, we see practical applications. The idea extends to advanced geometry topics, helping us understand structure and shape. It isn't just about static figures but the dynamic possibilities of creation and interaction within any geometric constraints.