Problem 7 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(A\) and \(B\) are the coefficient of \(x^{n}\) in the expansions of \((1+x)^{2 n}\) and \((1+x)^{2 n-1}\) respectively, then (a) \(A=B\) (b) \(A=2 B\) (c) \(2 A=B\) (d) none of these

Step-by-Step Solution

Answer

The correct answer is option (b) \(A = 2B\).

Step 1: Understanding the Problem Statement

We need to find the coefficient of \(x^n\) in the expansions of \((1+x)^{2n}\) and \((1+x)^{2n-1}\). Let's denote these coefficients as \(A\) and \(B\) respectively.

Step 2: Use of Binomial Theorem for Coefficient A

The binomial theorem states \((1+x)^m = \sum_{k=0}^{m}\binom{m}{k}x^k\). For \((1+x)^{2n}\), \(A\) is the coefficient of \(x^n\), which is \(\binom{2n}{n}\).

Step 3: Calculating Coefficient A

By substituting in the binomial coefficient formula, we find \(A = \binom{2n}{n}\).

Step 4: Use of Binomial Theorem for Coefficient B

Similarly, for \((1+x)^{2n-1}\), the coefficient \(B\) of \(x^n\) is \(\binom{2n-1}{n}\).

Step 5: Comparing A and B

Now we compare the expressions for \(A\) and \(B\). The relationship between \(\binom{2n}{n}\) and \(\binom{2n-1}{n}\) needs to be analyzed using binomial coefficient properties.

Step 6: Deriving the Relationship

The relation \(\binom{2n}{n} = 2 \times \binom{2n-1}{n}\) can be proven using the identity \(\binom{m}{k} = \binom{m-1}{k} + \binom{m-1}{k-1}\). By applying this identity, we can verify that \(A = 2B\).

Step 7: Final Conclusion

Through detailed analysis, it is concluded that \(A = 2B\). Hence, the correct option is (b) \(A = 2B\).

Key Concepts

Binomial Coefficient

The concept of binomial coefficients is central to understanding expansions in algebra. Binomial coefficients are the numbers that appear in the expansion of powers of a binomial expression. They are denoted as \( \binom{m}{k} \) and calculated using the formula:

• \( \binom{m}{k} = \frac{m!}{k!(m-k)!} \)

This formula gives the number of ways to choose \( k \) elements from a set of \( m \) elements. In simpler terms, it represents combinations in mathematics.
Each term in a binomial expansion features a coefficient which is one of these binomial coefficients. For example, in \((1+x)^m\), the coefficient of \(x^k\) is given by \(\binom{m}{k}\). Understanding how to calculate and apply these coefficients is essential for working with polynomial expansions.

Polynomial Expansion

In algebra, polynomial expansion allows us to express expressions like \((1+x)^n\) in an expanded form. The binomial theorem is key here, providing a formula to expand expressions raised to any power.
The expression \((1+x)^n\) expands to:

• \( \sum_{k=0}^{n} \binom{n}{k} x^k = 1 + nx + \frac{n(n-1)}{2!}x^2 + ... + x^n \)

This expansion reveals all the possible terms for the polynomial, with each term being a multiple of \(x^k\) and its corresponding binomial coefficient.
Understanding how polynomial expansion works means becoming familiar with the resulting terms and coefficients, especially when dealing with specific powers and identifying terms like \(x^n\). This process demonstrates combinatorial selection of elements and is widely used in combinatorics and other algebra applications.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and selection of objects. It plays a vital role when working with binomial coefficients and polynomial expansions.
At the heart of combinatorics concerning binomial coefficients, is the idea of choosing subsets from a larger set, commonly known as combinations.

• For instance, the binomial coefficient \( \binom{m}{k} \) represents the number of ways \( k \) items can be chosen from \( m \) items without considering the order of selection.

This concept is intrinsically linked to polynomial expansions like the binomial theorem, where each term in the expansion represents a particular combination.
In our specific problem, combinatorics is applied to determine how coefficients are derived, and why \(A\) (the coefficient of \(x^n\) in \((1+x)^{2n}\)) relates to \(B\) (the coefficient of \(x^n\) in \((1+x)^{2n-1}\)) as \(A = 2B\).

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols, and it plays a crucial part in solving polynomial expansions using the binomial theorem.
The problem at hand illustrates basic algebraic principles applied to higher-level mathematics, particularly how specific symbols (coefficients in this context) relate to others through algebraic rules.

• This involves understanding algebraic identities such as the identity: \( \binom{m}{k} = \binom{m-1}{k} + \binom{m-1}{k-1} \), which helps simplify and solve expressions.

By applying such identities, we derive relationships between coefficients in expansions, demonstrating the practical application of algebra in problem-solving.
Algebra simplifies the complexity of polynomial expansions, allowing us to derive conclusions like \(A = 2B\) efficiently and accurately. It's not only about handling numbers but understanding the underlying symbolic relationships and their application in various mathematical contexts.