Problem 46 - College Algebra

Question

In the expansion of \((a+b)^{n}\) , the coefficient of \(a^{n-k} b^{k}\) is the same as the coefficient of which other term?

Step-by-Step Solution

Answer

The coefficient of \(a^{n-k} b^{k}\) is the same as \(a^k b^{n-k}\).

Step 1: Recognize the Binomial Coefficient

In the expansion of \((a+b)^n\), each term is in the form \(C(n, k) \cdot a^{n-k}b^k\), where \(C(n, k)\) is the binomial coefficient \(\binom{n}{k}\).

Step 2: Find the Symmetric Term

The binomial coefficients have a property where \(\binom{n}{k} = \binom{n}{n-k}\). This means the coefficient of \(a^{n-k} b^{k}\) is the same as the coefficient of \(a^k b^{n-k}\).

Step 3: Identify the Term

Thus, in the expansion of \((a+b)^n\), the term \(a^{n-k} b^k\) has the same coefficient as the term \(a^k b^{n-k}\).

Key Concepts

Binomial Coefficient

When working with the expansion of expressions like \((a+b)^n\), the binomial coefficient plays a crucial role. It is denoted as \(C(n, k)\) or \(\binom{n}{k}\) and represents the number of ways to choose \(k\) elements from a set of \(n\) elements. This is calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n!\) denotes the factorial of \(n\). In the binomial expansion, each term is accompanied by a binomial coefficient, which determines its weight or size within the polynomial. It helps in systematically organizing terms so that you get the correct expansion without missing or adding any extra terms.

Polynomial Expansion

The polynomial expansion resulting from the binomial theorem is a structured way to write powers of a binomial expression \((a+b)^n\). In this expansion:

• Each term is a combination of two variables, \(a\) and \(b\), raised to different powers that together sum up to \(n\).
• The general term in the expansion is expressed as \(C(n, k) \cdot a^{n-k} b^k\).

The beauty of this expansion is how it breaks down complex expressions into manageable, smaller parts. For example, the expansion of \((a+b)^2\) gives three terms: \(a^2, 2ab,\) and \(b^2\). When expanded fully, these individual components provide a comprehensive representation of how the binomial theorem functions in algebraic manipulation.

Symmetry in Binomial Coefficients

One fascinating property of binomial coefficients is their symmetry. In mathematical terms, this symmetry is represented by the equation:\[ \binom{n}{k} = \binom{n}{n-k} \]This means that in the expansion of \((a+b)^n\), the term \(a^{n-k} b^k\) has the same coefficient as the term \(a^k b^{n-k}\). This symmetric property is useful for simplifying problems and predicting terms without having to explicitly calculate each one.
By recognizing symmetry, you can often find corresponding terms or identify the structure of an expression intuitively. This reflects an underlying balance in combinatorial mathematics, where the choices for \(k\) elements can be naturally mirrored by selecting their complements in a set of \(n\). The symmetry not only helps in verification but also in finding patterns in mathematical problems.