Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (A): \(\frac{n}{2}\).

Step 1: Understanding Combinations

In mathematics, the combination \(^{n}C_{r}\) represents the number of ways to choose \(r\) items from \(n\) items without regard to the order. Formally, it is given by \(^{n}C_{r} = \frac{n!}{r!(n-r)!}\). We want to find the value of \(r\) such that \(^{n}C_{r}\) is the largest value in this sequence of combinations.

Step 2: Pattern Analysis

Given in the problem is the pattern \({ }^{n} C_{0}<{ }^{n} C_{1}<{ }^{n} C_{2}<\ldots<{ }^{n} C_{r}>{ }^{n} C_{r+1}>\ldots>{ }^{n} C_{n}\). The combination values increase up to a certain point \(r\), and then start decreasing. To find \(r\), we need to analyze the nature of the combination function.

Step 3: Identifying the Peak in the Sequence

For combinations, the peak occurs at \(r\) when \(^nC_{r} = ^nC_{n-r}\) because they are symmetrical around \(n/2\). For \(n\) that is even, the sequence achieves its maximum value when \(r = \frac{n}{2}\), as the sequence is symmetric.

Step 4: Determine the Correct Answer

Given that the sequence of combinations increases up to \(r\) and then decreases afterwards, and knowing that \(r\) achieves its maximum value when \(r = \frac{n}{2}\) for an even \(n\), option \(A: \frac{n}{2}\) is the correct answer.

Key Concepts

Binomial Coefficients

In combinatorics, the concept of a binomial coefficient is central. Binomial coefficients, denoted as \( ^nC_r \) or sometimes \( \binom{n}{r} \), represent the number of ways to choose \( r \) items from a set of \( n \) items, where order does not matter. It is an essential part of the binomial theorem, used to expand expressions of the form \((a + b)^n\). These coefficients appear as constants in the expanded form.
An example of a binomial coefficient is \( ^5C_2 \), which evaluates to \( \frac{5!}{2!(5-2)!} = 10 \). This means there are 10 ways to pick 2 elements from a set of 5. Understanding this can simplify problems across statistics, probability, and algebraic expansions.

• The factorial function \(!\) is used in calculating binomial coefficients, where \(n!\) is the product of all positive integers up to \(n\).
• When calculating, consider using symmetry: \( ^nC_r = ^nC_{n-r} \), which greatly simplifies work when \( r \) exceeds \( \frac{n}{2} \).
• They form an important part of Pascal's Triangle, showcasing their utility in combinatorial problems.

Combination Formula

The combination formula is a fundamental expression in combinatorics that calculates the number of ways to select \( r \) elements from a larger set of \( n \) elements. It does not consider the order of selection, making it different from permutations. The formula is given by:
\[ ^nC_r = \frac{n!}{r!(n-r)!}.\]
Here, \( n! \) represents \( n \) factorial, which is the product of all positive integers up to \( n \). Similarly, \( r! \) and \( (n-r)! \) represent the factorial of \( r \) and \( n-r \), respectively.
The combination formula is potent in solving problems where the arrangement does not matter. For instance, selecting a committee of \( 3 \) members from \( 10 \) candidates is found using the formula as \( ^{10}C_3 \).

• The concept is used inherently in probability theory when calculating binomial probabilities.
• It eliminates duplicates, which is why it divides by \( r! \) to disregard the order.
• As combinatorics deals with counting possibilities, the combination formula is an efficient and concise tool.

Combinatorial Symmetry

Combinatorial symmetry is a fascinating property seen within combinatorics, particularly in binomial coefficients, where for a given \( n \), the coefficients are symmetrical about the middle. This means \( ^nC_r = ^nC_{n-r} \). This property greatly aids computations as it allows for simplification by choosing either \( r \) or \( n-r \), based on which is smaller.
This symmetry explains why, in a sequence of combinations, the values increase to a peak and then decrease symmetrically. For an even \( n \), the peak occurs at exactly \( r = \frac{n}{2} \), which is the middle of the sequence.

• This property can help in solving symmetry-related problems quickly, avoiding redundant calculations.
• Identifying the peak, especially in binomial distribution scenarios, leverages this symmetry to simplify solutions.
• In practical terms, knowing \( ^nC_r = ^nC_{n-r} \) helps confirm calculation accuracy as one expects matching values on either side of the sequence.