Problem 7 - Discrete Mathematics with Applications

Question

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$5 \cdot 4 !=5 !$$

Step-by-Step Solution

Answer

The given statement is \(\textbf{True}\), as evaluating both expressions, \(5 \cdot 4! = 120\) and \(5! = 120\), results in equal values.

Step 1: Understanding Factorial

Factorial is a mathematical operation, denoted as n! for a non-negative integer n, and is the product of all positive integers less than or equal to n. For example, \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\). It is important to note that by definition, \(0! = 1\).

Step 2: Evaluate and Compare the Expressions

In the given statement, we have two expressions: 5 ⋅ 4! and 5!. Evaluate both expressions: For \((5 \cdot 4!)\), we first need to determine the value of \(4!\), which is: \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\) Next, multiply this value by 5: \(5 \cdot 4! = 5 \cdot 24 = 120\) Second expression is \(5!\): \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5 \cdot (4 \cdot 3 \cdot 2 \cdot 1) = 5 \cdot 24 = 120\)

Step 3: Determine if the Statement is True or False

Compare the evaluated expressions: \(5 \cdot 4! = 120\) \(5! = 120\) Since both expressions are equal, the statement is True.

Key Concepts

Nonnegative integer

A nonnegative integer is any whole number that is not negative. This includes numbers like 0, 1, 2, and so on. Nonnegative integers are commonly used in mathematics, especially when dealing with counting or arranging objects. In various problems, we denote nonnegative integers with symbols such as $n$, indicating they cannot be less than zero. Examples of nonnegative integers are:

• 0
• 5
• 15
• 102

We often use nonnegative integers in problems that involve factorials, sequences, and other concepts that require a consistent set of positive values.

Factorial operation

The factorial operation is a fundamental concept in mathematics, especially in statistics, algebra, and calculus. It is denoted by an exclamation mark (!). If we write \(n!\), we mean the product of all positive integers from 1 to \(n\). For example, \(5!\) represents the product \(5 \times 4 \times 3 \times 2 \times 1\).
An important rule to remember is that the factorial of 0 is 1, or \(0! = 1\), by definition. This might seem odd initially, but it simplifies many mathematical formulas and expressions. Factorials grow very quickly, and their primary use is in permutations, combinations, and other areas where arrangements of numbers are essential.

Comparison of expressions

Comparing mathematical expressions is a skill where we determine if two quantities are equal or which one is greater or lesser. When expressions involve operations like factorials, the comparison typically involves calculating each side first.
For example, consider the expressions \(5 \times 4!\) and \(5!\). To determine if they are equal, start by evaluating both:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• Then, \(5 \times 24 = 120\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Both sides equal 120, so they are the same. Comparing expressions often requires understanding order of operations and the properties of numbers.

Mathematical operations

Mathematical operations such as addition, subtraction, multiplication, and division are the basics of mathematics. They help us solve problems, determine equations' solutions, and compare values. Factorials, like $n!$, represent a series of multiplications.
When performing operations with factorials, remember the following processes:

• Calculate the factorial by multiplying the sequence of decreasing numbers.
• Ensure you adhere to order of operations; factorial calculations should be completed before multiplication or addition with other terms.
• Factorials can simplify down to numbers that are easy to handle in further calculations, like $4! = 24$ for easier multiplication.

Understanding these operations ensures you can work smoothly with expressions involving factorials or other complex mathematical concepts.