Problem 80 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \({ }^{n} C_{r}\) denotes the number of combination of \(n\) things taken \(\mathrm{r}\) at a time, then the expression \({ }^{n} C_{r+1}+{ }^{n} C_{r-1}^{\prime}+2 \times{ }^{n} C_{r}\) equals \(\quad[2003]\) (a) \({ }^{n+1} C_{r+1}\) (b) \({ }^{n+2} C_{r}\) (c) \({ }^{n+2} C_{r+1}\) (d) \({ }^{n+1} C_{r}\)

Step-by-Step Solution

Answer

(c) \( { }^{n+2}C_{r+1} \).

Step 1: Understanding Combinations

The notation \( ^{n}C_{r} \) represents the number of combinations of \( n \) items taken \( r \) at a time, calculated using the formula \( ^{n}C_{r} = \frac{n!}{r!(n-r)!}\). We need to work with combinations in this problem.

Step 2: Identify the Problem Expression

The expression given is: \( { }^{n}C_{r+1} + { }^{n}C_{r-1} + 2 \times { }^{n}C_{r} \). We need to simplify this expression in terms of combinations.

Step 3: Applying Combination Identities

Use the known identity for combinations: \( ^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}\). Apply this identity to each part of the expression:- Start with \( { }^{n}C_{r+1} + { }^{n}C_{r} \). By identity, this equals \( { }^{n+1}C_{r+1} \).- Similarly, \( { }^{n}C_{r} + { }^{n}C_{r-1} = { }^{n+1}C_{r} \).

Step 4: Substituting into the Expression

Insert these simplified expressions into the original problem statement:- Start with \( { }^{n+1}C_{r+1} \) from the identity \( { }^{n}C_{r+1} + { }^{n}C_{r} \).- Add \( { }^{n+1}C_{r} \) for the combination \( { }^{n}C_{r} + { }^{n}C_{r-1} \).- The expression becomes \( { }^{n+1}C_{r+1} + { }^{n+1}C_{r} \).

Step 5: Final Combination Identity

Using another identity \( { }^{n+1}C_{r+1} + { }^{n+1}C_{r} = { }^{n+2}C_{r+1} \) (since adding combinations shifts \( n \) upward), we simplify the expression to match one of the answer choices.

Step 6: Select the Correct Answer

The simplified expression matches the identity \( { }^{n+2}C_{r+1} \). Therefore, the correct answer is (c) \( { }^{n+2}C_{r+1} \).

Key Concepts

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics. It represents the number of ways to select a certain number of items from a larger set, without regard to order. Typically denoted as \( { }^{n}C_{r} \) or sometimes \( \binom{n}{r} \), the binomial coefficient is calculated using the formula:

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

In this formula, \( n! \) ("n factorial") is the product of all positive integers up to \( n \), \( r! \) is the factorial of \( r \), and \( (n-r)! \) is the factorial for the difference between \( n \) and \( r \).
This formula allows us to calculate how many ways we can choose \( r \) objects from \( n \) objects. It's widely used in probability and statistics, and in combinatorial problems where the order of selection does not matter.

Combination Identities

Combination identities are equations that relate different binomial coefficients in combinatorics. These identities help simplify complex expressions involving multiple combinations.
One of the basic identities is:

• \( { }^{n}C_{r} + { }^{n}C_{r-1} = { }^{n+1}C_{r} \)

This identity explains how to sum two combinations; one with \( r \) items and one with \( r-1 \) items. It results in a binomial coefficient with \( n+1 \) items, still choosing \( r \). Another useful identity is:

• \( { }^{n+1}C_{r+1} + { }^{n+1}C_{r} = { }^{n+2}C_{r+1} \)

These identities are powerful tools for simplifying expressions and verifying equalities in combinatorial problems. They are especially handy in circumstances that involve evaluating sums over sequential binomial coefficients.

Combinatorial Mathematics

Combinatorial mathematics is an area of mathematics focused on counting, arrangement, and combination of elements within a set according to specified rules. It has diverse applications in fields such as computer science, statistical physics, and network theory, to name just a few.
At its core, combinatorics uses principles like:

• Counting Principle: Techniques for counting the number of ways to perform a task.
• Pigeonhole Principle: A simple yet powerful concept for proving that something exists based on counting.
• Inclusion-Exclusion Principle: This helps find the number of elements in the union of sets by subtracting overcounts from sum of individual counts.

Combinatorial mathematics overlaps with other branches such as algebra, graph theory, and probability, providing a framework for solving complex and fascinating problems.