Problem 46 - College Algebra with Corequisite Support

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 term is \(a^k b^{n-k}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), \((a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Each term in the expansion has a coefficient \(\binom{n}{k}\), where it corresponds to the term \(a^{n-k} b^{k}\).

Step 2: Recognize the Symmetry in Binomial Coefficients

The binomial coefficients are symmetric, which means \(\binom{n}{k} = \binom{n}{n-k}\). This symmetry means the coefficient for \(a^{n-k} b^k\) is the same as for \(a^k b^{n-k}\).

Step 3: Determine the Term with the Same Coefficient

Since the coefficients are symmetric, the term \(a^k b^{n-k}\) will have the same coefficient as \(a^{n-k} b^k\). This is derived from the property \(\binom{n}{k} = \binom{n}{n-k}\).

Key Concepts

Symmetry of Binomial Coefficients

The symmetry in binomial coefficients is a fascinating property that simplifies calculations significantly. When we talk about the symmetry of binomial coefficients, we refer to the fact that the coefficients in a binomial expansion, denoted as \( \binom{n}{k} \), are equal to \( \binom{n}{n-k} \). This tells us that every term in the expansion has a twin.
This means, for terms in the expansion of \((a+b)^n\), the coefficients associated with the term \(a^{n-k} b^k\) and the term \(a^k b^{n-k}\) are exactly the same.

• This property arises from the combinatorial interpretation of the binomial coefficients, where choosing \(k\) objects from \(n\) is the same as leaving \(k\) objects out, which is equivalent to choosing \(n-k\) objects.
• This symmetry is visually evident when you observe the well-known Pascal's Triangle, where each row is symmetrical about its center.

Understanding this symmetry helps in predicting the behavior of polynomial expansions without having to calculate each coefficient individually.

Expansion of Binomials

The expansion of binomials is a key application of the Binomial Theorem. The theorem provides a formula to write expressions of the form \((a+b)^n\) as a sum of terms involving powers of \(a\) and \(b\). The formula is:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\]This expansion consists of \((n+1)\) terms, and in each term, \(k\) represents the index of summation ranging from 0 to \(n\).

• The exponent of \(a\) decreases from \(n\) to 0 as \(k\) increases.
• Conversely, the exponent of \(b\) increases from 0 to \(n\).

Each term's coefficient \(\binom{n}{k}\) is a binomial coefficient representing the number of ways to choose \(k\) objects from \(n\). These concepts allow for the simplification of calculations in algebra and provide insight into the distribution and combination possibilities for various elements in computations.

Binomial Coefficients

Binomial coefficients are a central element in the study of algebraic expressions and combinatorics. Denoted \(\binom{n}{k}\), these coefficients indicate the number of ways to choose \(k\) elements out of \(n\) without taking the order into account, and they are prominently featured in the expansion of binomials using the Binomial Theorem.
Computationally, binomial coefficients can be derived using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, \(!\) stands for factorial, meaning the product of all positive integers up to that number.

• For example, \(\binom{5}{2} = \frac{5!}{2!3!} = 10\), which means there are 10 ways to choose 2 elements from a set of 5.
• They depict not only the coefficients in polynomial expansions but also the combinatorial selections in statistical analysis and probability theory.

Thus, binomial coefficients bridge the gap between algebra and the combinatorial ideas of counting and arranging, playing a crucial role in understanding and solving mathematical problems.