Problem 154 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(^{n} C_{r}\) denotes the number of combinations of \(n\) things taken \(r\) at a time, then the expression \({ }^{n} C_{r+1}+{ }^{n} C_{r-1}+\) \(2 \times{ }^{n} C_{r}\) equals (A) \({ }^{n+2} C_{r}\) (B) \({ }^{n+2} C_{r+1}\) (C) \({ }^{n+1} C_{r}\) (D) \({ }^{n+1} C_{r+1}\)

Step-by-Step Solution

Answer

The expression equals \(^{n+1} C_{r+1}\), so the correct option is (D).

Step 1: Understanding the Problem

We need to simplify the expression \(^{n} C_{r+1} + ^{n} C_{r-1} + 2 imes ^{n} C_{r}\) and find which of the given options it equals.

Step 2: Using Combination Properties

Recall that combinations have the identity: \(^{n} C_{r} = ^{n-1} C_{r} + ^{n-1} C_{r-1} \). This is useful for solving problems involving combinations by breaking them down into smaller parts.

Step 3: Apply the Combination Identity

Apply the identity to rewrite \(^{n} C_{r+1}\) and \(^{n} C_{r-1}\) in terms of combinations with lower indices and sums. For instance: \(^{n} C_{r+1} = ^{n-1} C_{r+1} + ^{n-1} C_{r}\) and \(^{n} C_{r-1} = ^{n-1} C_{r-1} + ^{n-1} C_{r-2}\).

Step 4: Substitute and Simplify

Substitute these expressions back into the original equation: \(^{n} C_{r+1} + ^{n} C_{r-1} + 2 imes ^{n} C_{r} = (^{n-1} C_{r+1} + ^{n-1} C_{r}) + (^{n-1} C_{r-1} + ^{n-1} C_{r-2}) + 2 imes (^{n-1} C_{r} + ^{n-1} C_{r-1})\).

Step 5: Combine Like Terms

Combine all like terms from the above expression. You will have multiple terms of \(^{n-1} C_{r}\) and \(^{n-1} C_{r-1}\) along with others. Simplify the expression according to combination identities, aiming to format it into a standard combination expression of type \(^{n+1} C_{k}\).

Step 6: Final Simplification and Result Comparison

After simplifying, the expression becomes \(^{n+1} C_{r+1}\). Compare this result to the options given to find the correct answer.

Key Concepts

Combinations

In mathematics, a combination is a way to select items from a larger pool, where the order in which they are picked does not matter. This concept is particularly useful when you want to find out how many different ways you can choose a certain number of items from a specific set. If you have ever wondered how many different ways you can choose 3 apples from a basket of 5, combinations will give you the answer.

Combinations are calculated using a formula involving factorials. Factorials are denoted by an exclamation mark (!) and represent the product of all positive integers up to a given number. The formula for combinations is:

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

Here, \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ^{n}C_{r} \) represents the number of possible combinations.

Combinations are distinct from permutations, where the order of selection does matter. For permutations, different orders of the same items are considered different selections. In combinations, they are seen as the same. Understanding this distinction is crucial when solving problems involving combinations.

Binomial Coefficients

Binomial coefficients are specific numbers that appear in the expansion of a binomial expression raised to a power. They have great importance in combinatorics and are widely used in algebra and probability. A binomial expression is made up of two terms, such as \((x + y)\), and when raised to a power \( n \), the coefficients of each term in its expansion are called binomial coefficients.

These coefficients correspond directly to the number of combinations of objects. The binomial coefficient for term \( r \) in the expansion is given by the formula:

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

This is exactly the same formula as that used for combinations, showcasing a direct connection between the two concepts.

The binomial theorem provides a way to calculate these coefficients, and it can be written as:

• \( (x + y)^n = \sum_{r=0}^{n} \, ^{n}C_{r} \, x^{n-r} \, y^{r} \)

In this expansion, each term is determined by a binomial coefficient, showing the power of binomial expressions to simplify complex algebraic formulas and calculations.

Mathematical Identities

Mathematical identities are equations that are true for all variable values within their range of definitions. They are powerful tools in simplifying mathematical expressions and solving equations. In the realm of combinatorics, one important identity is the combination identity used to simplify expressions and solve problems involving combinations.

This identity states:

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

It reflects how choosing \( r \) objects from \( n \) can be broken down into choosing \( r \) objects from \( n-1 \) plus choosing \( r-1 \) objects from \( n-1 \).

This identity is a fundamental part of simplifying complex combinations expressions. It allows us to express a combination involving higher numbers in terms of simpler combinations. The exercise mentioned utilizes this identity to find that:

• The expression \( ^{n}C_{r+1} + ^{n}C_{r-1} + 2 \times ^{n}C_{r} \) simplifies to \( ^{n+1}C_{r+1} \).

This demonstrates how mathematical identities can transform a complicated problem into a straightforward solution, revealing the beauty and power of mathematics in problem-solving.