Problem 6 - Discrete Mathematics with Applications

Question

Find the number of terms in the expansion of each expression. $$(a+b)(c+d+e)(x+y)$$

Step-by-Step Solution

Answer

There are 12 terms in the expansion of the expression \((a+b)(c+d+e)(x+y)\).

Step 1: Analyzing the numbers of terms in each factor

Let's start by analyzing each pairwise combination: 1. The first factor (a+b) has 2 terms: a and b. 2. The second factor (c+d+e) has 3 terms: c, d, and e. 3. The third factor (x+y) has 2 terms: x and y.

Step 2: Finding the total number of terms in the expansion

Now, let's find the total number of terms in the expansion by multiplying the number of terms in each factor together. Since each term from the first factor can be combined with each term from the second factor and each term from the third factor, we get: Number of terms in the expansion = (Number of terms in the first factor) × (Number of terms in the second factor) × (Number of terms in the third factor) Number of terms in the expansion = 2 × 3 × 2 = 12 So, there are 12 terms in the expansion of the given expression.

Key Concepts

Binomial Theorem

The Binomial Theorem presents a way to expand algebraic expressions that are raised to a power. While the exercise presented does not directly involve raising a binomial to a power, understanding the Binomial Theorem can help students recognize patterns in algebraic expansion. The theorem states that \( (a+b)^n \) can be expanded into a sum involving terms of the form \( a^r b^{n-r} \) multiplied by combinatorial coefficients. These coefficients correspond to the number of ways to choose \( r \) elements from \( n \) options and are represented by \( C(n, r) \), also known commonly as \( nCr \).

Take a simpler case such as \( (a+b)^2 \) which expands to \( a^2 + 2ab + b^2 \), where coefficients 1, 2, and 1 are the actual binomial coefficients for n=2. This concept is pivotal when dealing with more complex expressions involving multiple terms and factors, providing a foundation for understanding algebraic expansion on a broader scale.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like a, b, c), and operators (like addition and multiplication). When it comes to expanding these expressions, as seen in our exercise, the multiplication of terms across different factors is key. Each term from one factor must be multiplied by each term from another factor to find all possible products. For instance, the expression \( (a+b)(c+d+e)(x+y) \) results in each term from \( (a+b) \) being multiplied by each term from \( (c+d+e) \) and \( (x+y) \) to create the final expanded form. This illustrates the fundamental principle of distributive property in algebra—multiplying out and ‘distributing’ a term across others.

In the context of our example, a student must understand how to systematically multiply each term across factors to determine the resulting terms in the expansion, and that the number of terms in the final expression will be equal to the product of the number of terms in each factor.

Combinatorics

Combinatorics is a branch of mathematics focusing on the counting, arrangement, and combination of objects. In the realm of polynomial expansion, combinatorics involves determining the number of ways terms can be combined when expanding such expressions, much like the process of finding the number of distinct terms in the exercise. The term 'combinatorial coefficients' mentioned in the Binomial Theorem section are an example of combinatorial numbers in action.

When applying combinatorics to algebraic expansion, we can see how it determines the resulting number of terms. For the exercise \( (a+b)(c+d+e)(x+y) \), it’s not about raising to a power but rather about combining each term from one factor with each term from another. By counting the terms in each grouping and multiplying these counts, we are employing combinatorial principles to find the total number of terms. This kind of multiplicative counting is at the heart of combinatorial reasoning and showcases the importance of this branch of mathematics in understanding and solving algebraic problems.