Problem 27 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The coefficient of \(\lambda^{\pi} \mu^{n}\) in the expansion of \([(1+\lambda)(1+\mu)\) \((\lambda+\mu)]^{n}\) is (A) \(\sum_{r=0}^{n} C_{r}^{2}\) (B) \(\sum_{r=0}^{n} C_{r+2}^{2}\) (C) \(\sum_{r=0}^{n} C_{r+3}^{2}\) (D) \(\sum_{r=0}^{n} C_{r}^{3}\)

Step-by-Step Solution

Answer

The coefficient is \(\sum_{r=0}^{n} C_{r}^{2}\) (Option A).

Step 1: Understanding the Expansion

The given expression is \((1+\lambda)(1+\mu)(\lambda+\mu)^n\). We need to find the coefficient of \(\lambda^{\pi} \mu^{n}\) in this expansion.

Step 2: Expanding (λ+μ)^n

The term \((\lambda+\mu)^n\) can be expanded using the binomial theorem as: \((\lambda+\mu)^n = \sum_{r=0}^{n} \binom{n}{r} \lambda^r \mu^{n-r}\).

Step 3: Distributing (1+λ)(1+μ)

Distribute \((1+\lambda)(1+\mu)\) across the expanded terms: \((1+\lambda)(1+\mu)\sum_{r=0}^{n} \binom{n}{r} \lambda^r \mu^{n-r}\).

Step 4: Calculation of Coefficient

After distributing, select terms from \((1+\lambda)(1+\mu)\) and multiply corresponding terms from \((\lambda+\mu)^n\) to form terms. Collecting \(\lambda^\pi\mu^n\) gives the coefficient as: \(\sum_{r=0}^{n} \binom{r}{2} \). This is because we require \((\lambda\mu)^{n-r}\)to multiply with terms yielding \(\lambda^\pi\mu^n\).

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential mathematical tool for expanding expressions raised to a power. It provides a systematic way to expand expressions of the form \[(a+b)^n,\]as a sum of terms \[(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r. \] The coefficients that appear in the expansion are known as binomial coefficients and are represented by \[\binom{n}{r}\],which reads as "n choose r." These coefficients can be calculated using the formula:

• \[\binom{n}{r} = \frac{n!}{r!(n-r)!}.\]

This theorem applies not only to numbers but also to any variables within the expression. It is particularly useful in polynomial expansion, where complex expressions can be broken down into simpler individual terms. Expanding expressions using the Binomial Theorem simplifies calculations and helps form a deeper understanding of algebraic identities.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in a different form, usually as an expanded sum of individual terms. Given an expression raised to a power, such as \[(\lambda+\mu)^n,\]the goal is to rewrite it in terms of simpler polynomials. The expanded form consists of sums of terms, each represented by constants (coefficients) followed by variables raised to various powers.
For example, applying the Binomial Theorem, the expression \[(a+b)^n\]can be expanded into a polynomial with terms of the form \[\binom{n}{r}a^{n-r}b^r,\]which showcases the systematic distribution of the product terms. This approach helps clarify potential solutions to problems involving polynomial expressions by highlighting individual components that contribute to its overall structure.
Polynomial expansions are crucial for solving problems in mathematics, allowing you to see the relationships between terms, making computations more manageable.

Combinatorics

Combinatorics involves the study of counting, arrangement, and combination of elements within a set. It is essential in determining the number of possible ways to select or arrange items.

• In the context of polynomial expansion, combinatorics helps in identifying the appropriate binomial coefficients.
• The "n choose r" concept mirrors real-world scenarios where you need to select r items from a set of n items.

For example, in our expression, \[\binom{n}{r}\]can be interpreted as determining how many different ways we can choose r occurrences of \(\mu\)in a set of n tries.
Combinatorial techniques further extend and justify outcomes within algebraic operations. They provide a foundation for understanding the manipulation of variables and coefficients, which results in the expression reaching its expanded form through established mathematical principles.

Pascal's Triangle

Pascal's Triangle is a mathematical array of numbers arranged in a triangular format. It is closely tied to the binomial theorem, where the rows of the triangle correspond to the coefficients in the binomial expansion.

• Each number in Pascal's Triangle is the sum of the two directly above it in the previous row.
• The nth row corresponds to the coefficients of \((a+b)^{n-1}.\)

This triangular structure provides a convenient and easy way to identify binomial coefficients without directly computing factorials. For instance, in the expansion of \((\lambda+\mu)^n,\)Pascal's Triangle helps quickly locate coefficients like \[\binom{n}{r},\]which are necessary for determining specific terms in a polynomial.
Utilizing Pascal’s Triangle as a visual aid reinforces your understanding of how the expansion terms relate and simplifies calculations by employing patterns rather than complex arithmetic.