Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$3 ! \cdot 4$$

Step-by-Step Solution

Answer

24

Step 1: Understanding the Factorial Notation

The notation "\(!\)" represents a factorial. A factorial of a positive integer \(n\) is the product of all positive integers less than or equal to \(n\). Thus, \(3!\) means \(3 \times 2 \times 1\).

Step 2: Calculate the Factorial

Calculate \(3!\) by multiplying the integers: \(3 \times 2 \times 1 = 6\).

Step 3: Multiply the Factorial by 4

Now that we know \(3! = 6\), we multiply this result by 4: \(6 \times 4\).

Step 4: Final Calculation

Perform the multiplication: \(6 \times 4 = 24\).

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations that involves combining equal groups. In this exercise, you encountered multiplication when you calculated the factorial and then multiplied that by 4.
To multiply two numbers, think of adding one number to itself repeatedly. For example, multiplying 6 by 4 means adding 6, four times: 6 + 6 + 6 + 6, which equals 24.

• When multiplying, each number is often called a factor, and the result is called the product.
• Factors are important because they help you see how numbers can be grouped or divided in different ways.
• Multiplication is commutative, which means the order of multiplication does not affect the result: 6 × 4 gives the same product as 4 × 6.
• The multiplication we performed in this exercise confirms this rule, showing a solid base for understanding how factorials operate together with multiplication.

Integer operations

Integer operations are basic calculations involving whole numbers, and they are crucial for solving problems like this one involving factorials. An integer is any whole number, positive or negative, and zero.
In the exercise, we dealt exclusively with positive integers: 3, 2, 1, and 4.

• When calculating the factorial of 3, the operations 3 × 2 × 1 were performed entirely with positive integers.
• Operations involving integers follow specific rules: integers can be added, subtracted, multiplied, and divided—all following arithmetic principles.
• Here, multiplication of integers led to calculating the factorial before applying it further to another integer, which is a typical use-case for such operations.
• Using integer operations helps reinforce a clear understanding of how numbers interact, supporting more complex problem-solving techniques involving larger numbers.

Problem-solving skills

Problem-solving skills are essential in mathematics, as they enable you to approach and solve exercises by applying known concepts and strategies. In this exercise, we used such skills to evaluate the expression involving a factorial and multiplication.
Understanding factorial notation was the first step, showcasing critical thinking as you broke down the problem into manageable tasks.

• You recognized the notation and determined the correct multiplication sequence for 3!, which required attention to detail and a methodical approach.
• Once you calculated the factorial, you applied the result to a secondary multiplication with 4.
• This demonstrates sequential thinking, as you followed a series of steps or procedures to reach a conclusion.
• Developing problem-solving skills through practice with such exercises enhances your ability to quickly identify and execute steps needed in various mathematical tasks.
• Ultimately, honing these skills helps build confidence in tackling unfamiliar problems through logical reasoning.