Problem 47 - Discrete Mathematics with Applications

Question

Prove Vandermonde's identity algebraically. [Hint: Consider \((1+x)^{m}(x+1)^{n}=(1+x)^{m+n} .\)]

Step-by-Step Solution

Answer

To prove Vandermonde's identity algebraically, consider \((1+x)^m(1+x)^n = (1+x)^{m+n}\). Expand both sides using the binomial theorem: \[(1+x)^m = \sum_{k=0}^m \binom{m}{k} x^k\] and \[(1+x)^n = \sum_{j=0}^n \binom{n}{j} x^j.\] Multiply the two expansions and collect the terms with the same x-powers: \[\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} x^r\] Also, expand \((1+x)^{m+n}\): \[(1+x)^{m+n} = \sum_{p=0}^{m+n} \binom{m+n}{p} x^p\] Compare coefficients of like terms from both sides: \[\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}\] Thus, Vandermonde's identity is proved algebraically.

Step 1: Expand using binomial theorem

First, we need to expand \((1+x)^m(1+x)^n\) and \((1+x)^{m+n}\) using the binomial theorem, which states: \[(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k.\] For our specific case, expand \((1+x)^m\): \[(1+x)^m = \sum_{k=0}^m \binom{m}{k} (1)^{m-k} x^k = \sum_{k=0}^m \binom{m}{k} x^k.\] Similarly, expand \((1+x)^n\): \[(1+x)^n = \sum_{j=0}^n \binom{n}{j} x^j.\]

Step 2: Multiply the expansions

Now we need to multiply the two expansions. Notice that each term in the left-hand-side comes from multiplying one term from the first expression and one term from the second expression. \[(1+x)^m(1+x)^n = \left(\sum_{k=0}^m \binom{m}{k} x^k\right)\left(\sum_{j=0}^n \binom{n}{j} x^j\right)\]

Step 3: Collect terms and compare coefficients

After multiplying the terms, we need to collect the terms with the same x-powers and compare their coefficients. \[\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} x^r \] Now, expand \((1+x)^{m+n}\): \[(1+x)^{m+n} = \sum_{p=0}^{m+n} \binom{m+n}{p} x^p\]

Step 4: Compare both sides and conclude the proof

Now, we compare coefficients of like terms from both sides. The coefficients of x^r terms on both sides should be equal, that is: \[\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}\] Hence, we have proved Vandermonde's identity algebraically.

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that enables us to expand expressions raised to a power. Imagine you have an expression like (1 + x)^n\text{.} The binomial theorem tells us we can write this as a sum of several terms, where each term consists of coefficients from Pascal's triangle (known as binomial coefficients) and the variables raised to specific powers.

The general form of the binomial theorem for any real number n and any real numbers a and b is: \[(a + b)^n = \textstyle\biggl(\binom{n}{0} a^n b^0\biggr) + \textstyle\biggl(\binom{n}{1} a^{n-1} b^1\biggr) + \textstyle\biggl(\binom{n}{2} a^{n-2} b^2\biggr) + \textstyle\biggl(\binom{n}{k} a^{n-k} b^k\biggr) + ... + \textstyle\biggl(\binom{n}{n} a^0 b^n\biggr).\]
These coefficients, \( \binom{n}{k} \), are pronnounced 'n choose k' and represent the number of ways to choose k items from a set of n distinct items. This ties the theorem to combinatorial mathematics. For the Vandermonde's identity, we use the binomial theorem to expand both \((1 + x)^m\) and \((1 + x)^n\). Understanding this theorem is crucial for approaching algebraic problems involving powers of binomials.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and has applications in fields as diverse as computer science and statistical physics.

At the heart of combinatorics is the concept of the number of combinations, which calculate the number of ways a subset of items can be chosen from a larger set, without regard to the order of selection.
This is represented by the binomial coefficient notation \( \binom{n}{k} \), which tells us the number of ways to pick k elements out of a set of n distinct elements. Mathematically, it's given by: \[\binom{n}{k} = \frac{n!}{k!(n - k)!}\] where \( n! \) (n factorial) is the product of all positive integers up to n. This concept is inherently linked to the binomial theorem, as the coefficients in the binomial expansion are essentially combinatorial numbers. When applying combinatorics to algebraic expressions like Vandermonde's identity, we often use these coefficients to understand the multiplication and combinations of terms within expansions.

Algebraic Proof

An algebraic proof involves showing the validity of a mathematical statement by using algebraic manipulations and logically sequenced steps. The goal is to demonstrate that a statement holds true under given conditions. In the case of Vandermonde's identity, the algebraic proof requires transforming and comparing two different but equivalent expressions for \((1+x)^{m+n}\).

In the provided step-by-step solution, this proof process begins by using the binomial theorem to expand expressions. Following the expansion, algebraic techniques like distributing, collecting like terms, and comparing term coefficients are used. The strength of an algebraic proof lies in its structured approach—it breaks down complex problems into smaller, more manageable parts, each of which can be proved by standard algebraic rules.

For students, grasping the concept of an algebraic proof is essential not only for understanding specific identities like Vandermonde's but for tackling a wide range of problems in mathematics where direct evaluation isn't feasible.

Coefficient Comparison

Coefficient comparison is a technique used in algebra and calculus to determine if two polynomial expressions are equal. When two polynomials are equivalent, their corresponding coefficients for the same degree terms must be equal. This method is particularly useful when dealing with complex expansions from the binomial theorem.

In relation to Vandermonde's identity, the coefficient comparison is used in Step 4 of the solution. Both sides of the equation are expanded into polynomials, and the identity is verified by comparing the coefficients of corresponding powers of x. These comparisons lead us to conclude the proof because we can say with certainty that if all the corresponding coefficients match, then the two sides of the equation are identical.

This technique does not just validate identities; it also provides a framework for deriving new formulas or solving equations where direct calculation is not straightforward. Mastering coefficient comparison is a must for students delving into higher-level mathematics, as it facilitates understanding of the equivalence of polynomial expressions and functions.