Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ 2 \cdot 7 ! $$

Step-by-Step Solution

Answer

The given expression is \(2 \cdot 7!\). To evaluate this, we first find the factorial of 7: \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\). Then, we multiply this value by 2: \(2 \cdot 7! = 2 \cdot 5040 = 10080\). Therefore, the evaluated expression is \(10080\).

Step 1: Understanding Factorial (n!)

Factorial of a non-negative integer n (denoted as n!) is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Step 2: Calculate the Factorial of 7 (7!)

Using the definition of factorial, we can calculate 7! as follows: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Step 3: Calculate the Final Result

Now that we have the value of 7!, we can calculate the given expression: 2 ⋅ 7! First, we find the value of 7!: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 Next, we multiply the result by 2: 2 ⋅ 7! = 2 ⋅ 5040 = 10080 So, the value of the given expression is 10080.

Key Concepts

Factorial Calculation

Understanding the concept of factorials is a foundational element of mathematics, vital for complex problem-solving and various branches of the field, including combinatorics, algebra, and calculus. A factorial, written as a number followed by an exclamation point (n!), represents a product of an integer and all the non-zero integers below it.

For instance, the factorial of 5, denoted as 5!, is calculated as follows:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \].
In our exercise, the factorial of 7 (7!) is similarly calculated by multiplying 7 by every positive integer less than 7. So,
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \].
This process shows that understanding elaboration is crucial to accurately compute factorial values, ultimately aiding in solving more advanced problems.

Improving factorial calculations can be achieved through consistent practice, using factorials in various expressions, and familiarizing oneself with properties such as zero factorial being 1 (0! = 1).

Applied Mathematics

Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. This discipline thus involves both the pure mathematics of proof and abstraction and the practical ability to apply the principles to solve real-world problems.

Real-World Application of Factorials

Factorials, specifically, play an essential role in various applied mathematics contexts, such as in calculations of permutations and combinations, which are crucial in fields like data analysis, cryptology, and even in the formulation of algorithms.

The evaluation of the expression in the exercise, \(2 \cdot 7!\), can be seen as a simplified model of applied problems where similar principles would be required. For example, in probability theory, understanding factorials allows one to determine the number of possible outcomes in a lottery draw or the arrangement of books on a shelf.
Such applications underscore not only the importance of mastery but also of recognizing the relevance of mathematical operations in practical scenarios.

Problem-solving in Mathematics

Problem-solving stands at the heart of mathematics; it involves understanding the problem, devising a plan to solve it, carrying out the plan, and then looking back to review the solution and the process.

Strategic Approach

A strategic approach to problem-solving in mathematics includes breaking down complex problems into manageable parts, as was done in the step-by-step solution of the factorial problem.

Firstly, identifying that the problem requires the computation of factorial is part of problem recognition. Then, calculating 7! by progressively multiplying integers downwards from 7, and finally multiplying the computed factorial by 2 leads to the solution. This systematic process is important not just for factorials, but for solving various types of mathematical problems.
Learning and practicing such structured methods of reasoning can enhance one’s problem-solving skills, which are vital in both academic and real-world situations. Whether approaching algebraic equations, geometric proofs, or statistical analyses, the core problem-solving steps can guide students to clear, logical solutions.