Problem 46 - College Algebra

Question

In the expansion of \((a+b)^{n},\) the coeffici \(\quad \mathrm{t}\) of \(a^{n-k} b^{k}\) is the same as the coeffici \(\mathrm{t}\) 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: Understanding the Binomial Theorem

The binomial theorem states that the expansion of \((a+b)^n\) is given by \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient, representing the number of ways to choose \(k\) successes from \(n\) trials.

Step 2: Identifying the Desired Coefficient

We are interested in the term \(a^{n-k}b^{k}\) where the coefficient is given by \(\binom{n}{k}\). We want to find another term in the expansion with the same coefficient \(\binom{n}{k}\).

Step 3: Leveraging Symmetry in Binomial Coefficients

In the binomial expansion, due to symmetry, the coefficient of \(a^{n-k}b^{k}\) is the same as the coefficient of \(a^{k}b^{n-k}\). This is because \(\binom{n}{k} = \binom{n}{n-k}\).

Step 4: Finding the Equivalent Term

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

Key Concepts

Binomial Coefficient

In mathematics, the binomial coefficient is a key component of the binomial theorem. It is represented by the symbol \(\binom{n}{k}\), pronounced as "n choose k." This notation signifies the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection.
A crucial property of binomial coefficients is that they are symmetrical, meaning \(\binom{n}{k} = \binom{n}{n-k}\). This symmetry plays an essential role in understanding the concept within the binomial theorem, especially when exploring identities and expansions. Binomial coefficients can be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(!\) denotes factorial, the product of all positive integers up to a given number. These coefficients are used extensively in probability, statistics, and combinatorics as they represent combinations in many different scenarios.

Symmetry in Binomial Expansion

The symmetry in binomial expansion is a fascinating aspect of the binomial theorem that allows for elegant simplifications and deeper insights into polynomial expressions. The binomial theorem describes how the powers of a binomial, such as \((a + b)^{n}\), can be expanded and expressed as a sum of terms.
In this expansion, each term is a product of a binomial coefficient and powers of the individual binomial terms \(a\) and \(b\). A striking feature of binomial expansions is their symmetry. Specifically, the coefficient of a term \(a^{n-k}b^{k}\) is identical to the coefficient of the reversed term \(a^{k}b^{n-k}\). This is due to the property \(\binom{n}{k} = \binom{n}{n-k}\).
Understanding this symmetry can be extremely helpful in identifying equivalent terms in an expansion and simplifying complex algebraic expressions.

Combinatorial Identities

Combinatorial identities often emerge from the rich interactions between combinatorial structures and algebraic formulas. These identities frequently involve binomial coefficients and can be used to prove relationships and simplify expressions.
Examples of combinatorial identities include:

• Pascal's Identity: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\). This identity arises from considering the recursive structure of choosing elements.
• Symmetry Identity: As noted before, \(\binom{n}{k} = \binom{n}{n-k}\) is a foundational identity showcasing symmetry.
• Sum of Binomial Coefficients: \(\sum_{k=0}^{n} \binom{n}{k} = 2^{n}\), expressing a connection to the total number of subsets of a set with \(n\) elements.

Combinatorial identities are valuable in various fields, from simplifying algebraic expressions to modelling probabilistic scenarios and solving complex counting problems. These identities illustrate the beauty and utility of combinatorics in mathematics.