Problem 85 - A Complete Resource Book in Mathematics for JEE Main

Question

\({ }^{2 n} \mathrm{C}_{r}(0 \leq r \leq 2 n)\) is greatest when \(r\) is equal to (A) \(\frac{n}{2}\) (B) \(\frac{n+1}{2}\) (C) \(r=n\) (D) None of these

Step-by-Step Solution

Answer

The greatest value occurs at \(r=n\), so the answer is (C).

Step 1: Understand the Problem

The problem is about finding the value of \(r\) that maximizes the binomial coefficient \(^{2n}C_r\) for \(0 \leq r \leq 2n\). The binomial coefficient \(^{2n}C_r\) represents the number of combinations of \(2n\) items taken \(r\) at a time, which is maximum around the midpoint of the binomial distribution.

Step 2: Identify the Maximum Point of the Binomial Coefficient

For a binomial coefficient \(^{n}C_r\), it is well-known that it attains its maximum value at \(r = \lfloor \frac{n}{2} \rfloor\) or \(\lceil \frac{n}{2} \rceil\). This can be applied here considering \(n\) is replaced by \(2n\). The maximum occurs closest to \(r = \frac{2n}{2} = n\).

Step 3: Evaluate Given Options

Now compare the options provided with \(r=n\) since we've established that this is where the binomial coefficient reaches its peak for \(^{2n}C_r\). The option that matches is (C) \(r=n\).

Key Concepts

Combinations

In mathematics, combinations are a way to select items without considering the order. This concept is commonly used to determine the number of ways to choose a subset of elements from a larger set. Imagine you have a basket of 5 different fruits, and you want to pick 2. How many ways can you do this? This is where combinations come into play.

The formula for combinations is given by the binomial coefficient, expressed as \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to be chosen. This is calculated using the formula:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
Here, \(!\) represents the factorial, a product of an integer and all the integers below it.

• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• If you select \(r\) without regard to order, the number of possible combinations is computed using this formula.

This concept is particularly important in probability and statistics, where understanding the number of possible outcomes forms the basis of probabilistic calculations.

Maximum Binomial Coefficient

The binomial coefficient \(\binom{n}{r}\) describes the number of ways to choose \(r\) items from \(n\) items without regard to order. The value of this coefficient varies depending on \(r\), reaching its maximum at a specific point. To find where this maximum occurs, it's useful to understand its distribution: the binomial coefficient tends to peak at the middle.

For any given \(n\), \(\binom{n}{r}\) achieves its maximum when \(r\) is around half of \(n\). This happens because the distribution is symmetric, and the middle values cover the broadest array of combinations. Specifically, for a coefficient \(\binom{2n}{r}\):

• The maximum value generally occurs when \(r = n\).
• In cases where \(2n\) is an even number, the peak is exactly at this middle point.

This insight is useful because it helps to simplify the search for the maximum coefficient rather than calculating each individual value. In our original problem, the choice \(r = n\) was shown to be where \(\binom{2n}{r}\) peaks, thus confirming it as the correct answer.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with countable, distinct elements. Unlike continuous mathematics, which deals with smooth curves and calculations over an interval, discrete mathematics focuses on distinct, separate elements. This branch includes several fundamental concepts: sets, graphs, logic, and, pertinent to our discussion, combinations and permutations.

Discrete mathematics is essential in computer science for data structures and algorithms, underpinning many computational tasks. It's also critical in cryptography, network design, and computer architecture.

Combinations, as discussed, are a core part of discrete mathematics. They allow us to understand how elements can be selected and arranged without regard to sequence.

• These concepts are fundamental in algorithms that require counting arrangements, such as sorting curricula, assigning jobs, and optimizing paths.

Study in discrete mathematics gives foundational tools to approach problems in logical, structured ways. Understanding combinations and their properties, like the maximum binomial coefficient, provides essential insights not only in mathematics but in real-world problem-solving.