Problem 68 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The value of \(\mathrm{r}\) for which \({ }^{20} C_{r}{\underline{\phantom{xx}}}^{20} C_{0}+{ }^{20} C_{r-1}{\underline{\phantom{xx}}}^{20} C_{1}+{ }^{20} C_{r-2}{\underline{\phantom{xx}}}^{20} C_{2}+\ldots+{ }^{20} C_{0}{\underline{\phantom{xx}}}^{20} C_{r}\) ismaximum, is: (a) 15 (b) 20 (c) 11 (d) 10

Step-by-Step Solution

Answer

The value of \( r \) for which the sum is maximized is 10.

Step 1: Comprehend the Given Problem

We need to find the value of \( r \) such that the given sum, \( { }^{20} C_{r}{\underline{\phantom{xx}}}^{20} C_{0}+{ }^{20} C_{r-1}{\underline{\phantom{xx}}}^{20} C_{1}+{ }^{20} C_{r-2}{\underline{\phantom{xx}}}^{20} C_{2}+ ext{...}+{ }^{20} C_{0}{\underline{\phantom{xx}}}^{20} C_{r} \), is maximized. This expression is understood as the sum of terms involving binomial coefficients, where terms are symmetric around the midpoint of a complete binomial expansion.

Step 2: Identify Nature of the Sum

The problem essentially transforms the sum of these binomial terms into a specific point from the expansion of a binomial coefficient. Specifically, the sum resembles the identity involving coefficients from the expansion of the square of a binomial, \( \left(\sum_{i=0}^{20}{\underline{\phantom{xx}}}^{20}C_{i} x^i\right)^2\), at the midpoint, i.e., \( x^r \).

Step 3: Analyze Maximum Value in a Binomial Expansion

Since we have a symmetric problem involving binomial coefficients, the sum requires symmetry found at the middle. For maximum value in the distribution of a binomial sequence, the peak lies at \( r = \lfloor \frac{n}{2} \rfloor \) or \( \lceil \frac{n}{2} \rceil \). For \( n = 20 \), the maximum occurs at \( r = 10 \).

Step 4: Verify the Location of Maximum

To ensure the accuracy of the maximum occurrence, note that the central term (the highest sum) corresponds to \( r = 10 \) due to the properties of binomial coefficients distributing symmetrically about their midpoint.

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting, arrangement, and combination of objects. It forms the basis for solving problems involving selections and groupings of discrete items. In the context of our exercise, combinatorics provides the framework for understanding binomial coefficients. These coefficients, denoted as \( ^nC_r \), represent the number of ways to choose \( r \) objects from a set of \( n \) objects without regard to order.

• Combinatorial problems often involve calculating such coefficients to find the number of possible combinations.
• These coefficients appear in various mathematical scenarios, such as probability, statistics, and algebra.

Therefore, gaining a strong understanding of combinatorics can help simplify and solve many real-life problems that involve decision making through various combinations. In the given exercise, combinatorics helps determine the number of symmetrical ways we can choose coefficients in a binomial expansion.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions raised to a power, specifically the binomial expression \((a+b)^n\). The expansion is governed by the binomial theorem, which provides a formula for expressing these terms.

In the binomial expansion, each term is expressed as a binomial coefficient multiplied by the powers of \(a\) and \(b\). The coefficients \(^nC_0, ^nC_1, ..., ^nC_n\) are directly tied to combinatorics and represent the different ways to choose elements from a set.

• In our exercise, the expression is effectively related to the square of such an expansion.
• Understanding how these terms increase and decrease in a predictable pattern is central to solving the problem.

Thus, the binomial expansion provides both the structure and numerical values needed to analyze symmetric sums like those in this problem.

Symmetric Property of Binomial Coefficients

The symmetric property of binomial coefficients is a key principle that simplifies analyzing binomial expressions. This property states that \(^nC_r\) is equal to \(^nC_{n-r}\). This means the binomial coefficients are symmetrical around the midpoint of the expansion.

• This symmetry is crucial for identifying the highest value in a binomial expansion, as it points to the middle terms as the likely location.
• In our exercise, this symmetry helps us find that the center of the symmetry (where the maximum coefficient occurs for \(n=20\)) is at \(r=10\).

The symmetric property not only helps determine locations for maximum values but also aids in simplifying complex expressions. By understanding this, students can more easily navigate combinatorial problems that involve binomial coefficients.