Problem 59 - Algebra and Trigonometry

Question

Explain how to evaluate \(\left(\begin{array}{l}{n} \\ {r}\end{array}\right) .\) Provide an example with your explanation.

Step-by-Step Solution

Answer

The binomial coefficient of \( n=5 \) and \( r=3 \) is 10.

Step 1: Understanding the Binomial Coefficient

The binomial coefficient is the number of ways to choose \(r\) elements from a set of \(n\) elements. It is represented using the formula \( \left(\begin{array}{l}{n} \ {r}\end{array}\right) = \frac{n!}{r!(n-r)!} \), where \( n! \) denotes the factorial of \( n \) — the product of all positive integers up to \( n \).

Step 2: Evaluating an Example

Let's evaluate the binomial coefficient for \( n=5 \) and \( r=3 \). Substitute these values into the binomial coefficient formula: \( \left(\begin{array}{l}{5} \ {3}\end{array}\right) = \frac{5!}{3!(5-3)!} \).

Step 3: Calculating Factorials

Calculate the factorials for \( n \), \( r \), and \( n-r \). Here, \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120 \), \( 3! = 3 \cdot 2 \cdot 1 = 6 \), and \( 2! = 2 \cdot 1=2 \). So, our equation becomes \( \left(\begin{array}{l}{5} \ {3}\end{array}\right) = \frac{120}{6 \cdot 2} \).

Step 4: Simplification and Result

Simplify the expression to get the final result. The simplification gives us \( \left(\begin{array}{l}{5} \ {3}\end{array}\right) = \frac{120}{12} = 10 \). So, there are 10 ways to pick 3 elements from a set of 5.

Key Concepts

Factorials in Algebra

The concept of factorials is fundamental in algebra and serves as the cornerstone for various mathematical principles, including combinatorics and the evaluation of binomial coefficients. In essence, the factorial of a non-negative integer, denoted as  \( n! \), is the product of all positive integers less than or equal to  \( n \). For example,  \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It's crucial to note that  \( 0! = 1 \), a definition that simplifies many mathematical formulas.

Understanding factorials enables learners to manipulate algebraic expressions, develop combinatorial identities, and solve problems involving permutations and combinations. In problems requiring the evaluation of binomial coefficients, factorials allow one to count the number of ways in which items can be arranged or selected, emphasizing their importance in the broader concept of combinatorics.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results. It revolves around the concepts of permutations and combinations, which are methods of arranging or selecting items without and with regard to order, respectively. The fundamental principle upon which combinatorics is built is that of choice from a set — a concept perfectly encapsulated in the binomial coefficient.

For example, if you want to determine how many ways you can select three books from a shelf of five, you're delving into combinatorial analysis. In this context, the selected books don't need to be in a certain order; what matters is the act of choosing itself. Thus, combinatorial mathematics feeds into probability, optimization, and many other fields, providing us with the tools to analyze possibilities and make informed decisions based on counts.

Evaluating Binomial Coefficients

Evaluating binomial coefficients is a process that frequently appears in probability, statistics, and algebra. The binomial coefficient, represented as  \( \left(\begin{array}{l}{n} \ {r}\end{array}\right) \), determines the number of ways to choose  \( r \) items from a larger set of  \( n \) items, regardless of order. The formula for computing this coefficient is  \( \frac{n!}{r!(n-r)!} \).

When it comes to actually computing a binomial coefficient, start by calculating the factorials of  \( n \),  \( r \), and  \( n-r \), as shown in the example from the textbook. Simplification involves reducing the factorial expressions and dividing them accordingly, ultimately rendering the total number of combinations for the given values of  \( n \) and  \( r \). The process reflects not just an ability to manipulate numbers, but an understanding of how we count and classify distinct arrangements from a set—an important aspect of algebraic thinking and problem-solving.