Problem 29 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(n\) is even and \({ }^{n} C_{0}<{ }^{n} C_{1}<{ }^{n} C_{2}<\ldots\left\langle{ }^{n} C_{r}>{ }^{n} C_{r+1}>\ldots>{ }^{n} C_{n}\right.\) then \(r=\) (A) \(\frac{n}{2}\) (B) \(\frac{n-1}{2}\) (C) \(\frac{n-2}{2}\) (C) \(\frac{n+2}{2}\)

Step-by-Step Solution

Answer

The correct answer is (A) \(\frac{n}{2}\).

Step 1: Understand the Problem

We need to find the value of \(r\) for which the binomial coefficient \({ }^{n} C_{r}\) is maximum given the condition \({ }^{n} C_{0}<{ }^{n} C_{1}<{ }^{n} C_{2}<\ldots\). This happens when \(n\) is an even number.

Step 2: Analyze the Properties of Binomial Coefficients

Binomial coefficients, also called combinations, are expressed as \({ }^{n} C_{r} = \frac{n!}{r!(n-r)!}\). For a given \(n\), these coefficients increase until the middle of the binomial expansion and then start decreasing.

Step 3: Determine the Maximum Point for Even \(n\)

Since \(n\) is even and the binomial coefficients increase to a maximum point before decreasing, when \(n\) is even, the maximum coefficient occurs at the middle of the expansion. Thus, the maximum value comes at \(r = \frac{n}{2}\).

Step 4: Choose the Correct Option

Given the choices, the correct option that corresponds to \(r = \frac{n}{2}\) is option (A).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics focused on counting, arranging, and analyzing finite sets of elements. It's the study of how to count different configurations and is pivotal in various fields like computer science, statistics, and probability. A fundamental concept in combinatorics is the idea of combinations, which allows us to determine how many ways we can pick items from a larger set when the order doesn't matter.

A common notation used in combinatorics is the binomial coefficient, represented as \({ }^{n} C_{r}\), which calculates the number of ways to choose \(r\) items from a set of \(n\) items. The formula for this is:

• \(^{n} C_{r} = \frac{n!}{r!(n-r)!}\)

Understanding binomial coefficients is crucial, as they often describe the probabilities in scenarios where a choice needs to be made from a set of possibilities. This mathematical tool is essential not just for theoretical math problems but also for practical applications dealing with discrete structures and decision-making processes.

Pascal's Triangle

Pascal's Triangle is a simple yet powerful tool in combinatorics. It's a triangular array of numbers where each number is the sum of the two directly above it. This structure visually represents binomial coefficients. Each row in Pascal's Triangle corresponds to the coefficients in the binomial expansion of \((a+b)^n\).

For example, the third row (starting from row zero) of Pascal's Triangle is \(1, 2, 1\), which represent the coefficients \({ }^{2} C_{0}, { }^{2} C_{1}, { }^{2} C_{2}\). This gives us the expanded form of \((a+b)^2\). As you progress down the triangle, you discover patterns, like how numbers in a row always increase up to the middle, then reflect symmetrically.

Pascal's Triangle is not only an excellent tool to quickly find combinations and coefficients, but it also unveils deeper mathematical patterns and properties. It's utilized to simplify problems in binomial expansions and to demonstrate recursive relationships in algebra.

Mathematical Induction

Mathematical induction is a technique used to prove mathematical statements that are asserted to be true for all natural numbers. The process involves two primary steps: establishing a base case and then proving that if it's true for one case, it is true for the next.

Let's say you have a statement involving a natural number \(n\). First, you verify the statement for the initial value, usually \(n = 1\) or \(n = 0\), depending on the context. This is known as the base case. The next part, called the inductive step, involves showing that if the statement holds for some number \(k\), then it also holds for \(k+1\).

The principle of induction is like a domino effect. By toppling the first domino (the base case), and proving that each domino will knock down the next (the inductive step), you can conclude that all dominoes will fall, demonstrating the truth of the statement for all natural numbers. This powerful method is widely used to prove properties of numbers, sequences, and combinatorial objects effectively.