Problem 32 - Discrete Mathematics with Applications

Question

Using the binomial theorem, prove each. \(\sum_{r=0}^{n}\left(\begin{array}{c}n \\\ r\end{array}\right)\left(\begin{array}{c}n \\\ n-r\end{array}\right)\left(\begin{array}{c}2 n \\ n\end{array}\right)\) [Hint: Consider \((1+x)^{2 n}=(1+x)^{n}(1+x)^{n} .\) Equate the coefficients of \(\left.x^{n} \text { from either side. }\right]\)

Step-by-Step Solution

Answer

Here's the short answer: Using the hint, consider \((1+x)^{2n}=(1+x)^n(1+x)^n\). Expanding both sides using the binomial theorem gives: \((1 + x)^{2n} = \sum_{k=0}^{2n}\binom{2n}{k} x^k\) \((1+x)^n(1+x)^n = \left(\sum_{m=0}^n \binom{n}{m} x^m\right) \left(\sum_{m=0}^n \binom{n}{m} x^m\right)\) To find the coefficients of the \(x^n\) term, we equate the coefficients of \(x^n\) from both sides: \(\binom{2n}{n} = \sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r}\) This proves the identity: \(\sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r} = \binom{2n}{n}\)

Step 1: Rewrite the Equation

Let's rewrite the equation provided in the hint: \((1+x)^{2n}=(1+x)^n(1+x)^n\)

Step 2: Expand the Binomials

Now, we will expand both sides using the binomial theorem: The left side is: \((1 + x)^{2n} = \sum_{k=0}^{2n}\binom{2n}{k} x^k\) The right side is: \((1+x)^n(1+x)^n = \left(\sum_{m=0}^n \binom{n}{m} x^m\right) \left(\sum_{m=0}^n \binom{n}{m} x^m\right)\)

Step 3: Find the coefficient of \(x^n\) term

From the left side, the coefficient of \(x^n\) term is: \(\binom{2n}{n}\) From the right side, we need to multiply the above two summations. To find the coefficients of the \(x^n\) term, we need to find the combinations of terms (from the two summations) that result in \(x^n\). Elements that contribute to the \(x^n\) term are those elements with exponents summing up to \(n\). Thus, for the coefficients of the \(x^n\) term: \(\sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r}\)

Step 4: Equate coefficients and prove the identity

Now, we equate the coefficients of \(x^n\) term from both sides: \(\binom{2n}{n} = \sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r}\) This proves the given identity: \(\sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r} = \binom{2n}{n}\)

Key Concepts

Understanding Binomial Coefficients

The concept of a binomial coefficient is central to the binomial theorem. Binomial coefficients are represented as \( \binom{n}{r} \), which are often read as "n choose r." It determines the number of ways to choose \( r \) elements from a set of \( n \) elements without regard to the order of selection.
These coefficients are found in Pascal's triangle, where each number is the sum of the two numbers directly above it. In terms of formula, it is expressed as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \(!\) denotes factorial, a product of all positive integers up to a given number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Binomial coefficients are key in expanding powers of binomials, such as \((1+x)^n\), where each term is multiplied by the corresponding binomial coefficient, illustrating the different combinations possible.

Exploring Combinatorial Identities

A combinatorial identity is an equation involving sums and products of binomial coefficients, which holds true for any values fulfilling the equation’s constraints. The identity shown in the original exercise is a classic example:
\[ \sum_{r=0}^{n} \binom{n}{r} \binom{n}{n-r} = \binom{2n}{n} \]
This identity is essentially equating two different approaches to counting the same set of combinations. The left side calculates the total through permutations in pairs, while the right compresses the combinatory results into a single term, \(\binom{2n}{n}\), matching the card counting perspective.
In practical applications, recognizing and proving such identities can simplify complex counting problems effectively. The goal is often to show that multiple independent paths can lead to the same result.

The Role of Mathematical Proofs in Validation

Mathematical proofs provide a logical framework to verify that a theory or identity is valid. In the context of the binomial theorem and related identities, proofs are necessary to ensure the relationships are true for all possible values within set constraints.
Proofs can take various forms, such as direct proofs, inductive proofs, or combinatorial proofs. In this exercise, equating coefficients of \(x^n\) terms from both sides validates the identity via direct comparison.
Each proof step acts as a building block towards clarity, ensuring both sides of a statement reflect the same truth.

• Step 1: Rewriting equations to a common base.
• Step 2: Expanding using known formulas.
• Step 3: Matching specific terms for comparison.
• Step 4: Equating results to reach a conclusion.

Through these steps, the mathematical community confirms identities, fostering trust in mathematical reasoning.

Power Series Expansion in Binomial Theorem

Power series expansion involves breaking down expressions into an infinite sum, enabling precise calculus operation apart from simple polynomial operations.
In the binomial theorem context, \((1+x)^n\) is expanded as a series:
\[ \sum_{k=0}^n \binom{n}{k} x^k \]
For any integer \(n\), this expansion provides a method of expressing powers of an addition operation (both \(1+x\)) as a finite polynomial, with each term represented by a product of the coefficient and variable power.
It translates complex multiplicative expressions into manageable series computations, aiding in precise calculation and understanding.
The breakdown not only simplifies complex algebraic manipulations but allows clear identification of individual term impacts within the overall expansion. This deep insight into polynomial behavior is invaluable in areas like calculus, sequence summation, and even applied fields such as physics and engineering.