Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the factorial. \(6 !\)

Step-by-Step Solution

Answer

The factorial of 6, denoted as \(6!\), is 720.

Step 1: Understand Factorial Notation

Factorial of a number, denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(6!\) means the multiplication of all numbers from 1 to 6.

Step 2: Set Up the Expression

Write down the numbers from 1 to 6 that need to be multiplied together: \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 3: Perform the Multiplication

Perform the multiplication step-by-step: 1. \(6 \times 5 = 30\) 2. \(30 \times 4 = 120\) 3. \(120 \times 3 = 360\) 4. \(360 \times 2 = 720\) 5. \(720 \times 1 = 720\).

Step 4: Write the Final Answer

After performing all the multiplications, the result of \(6!\) is 720.

Key Concepts

Algebra

Algebra is an essential part of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, we often work with variables that represent numbers in equations and expressions.
Variables are symbols like \(x\) or \(n\) that stand for unknown values. When we evaluate factorials, such as \(6!\), we are essentially working with an algebraic expression. In this case, "6" is a constant, not a variable, but the process of simplifying expressions applies.

• Factorial notation involves multiplying a sequence of descending natural numbers, which introduces basic algebraic skills.
• Factorials are used in different areas of algebra, especially when dealing with combinatorial equations and series.
• Understanding factorials also helps break down complex algebraic expressions into simpler forms.

Grasping these algebraic principles is crucial as you advance, enabling easier manipulation and evaluation of mathematical expressions.

Integer Multiplication

Integer multiplication is the repeated addition of whole numbers. It forms the core operation when calculating factorials.
The factorial \(6!\) requires you to multiply the integers from 1 to 6, following a specific order:

• First, multiply 6 by 5, resulting in 30.
• Next, multiply 30 by 4 to get 120.
• This process continues, multiplying 120 by 3 (yielding 360), then multiplying 360 by 2, and finally by 1.

During each step, you maintain the integrity of multiplication by ensuring no number is left out or skipped over. This careful sequence is vital when converting factorial expressions into a single product.
When learning about integer multiplication in factorials, consider using visual aids, or breaking down each multiplication step to build confidence and accuracy.

Combinatorics

Combinatorics is a fascinating branch of mathematics focused on counting, arranging, and finding patterns. Factorials are integral to understanding combinatorial problems. This is because factorials represent the total number of ways to arrange a set of objects.
For example, calculating \(n!\) helps determine permutations, which are arrangements where order matters.

• The concept of \(6!\) tells us there are 720 unique ways to arrange six distinct items.
• Factorials also help compute combinations, where the order doesn’t matter, by dividing the result by additional factorials.

Understanding how and why factorials apply in combinatorics aids in solving various practical problems in fields such as probability, statistics, and computer science.
This knowledge extends beyond theoretical calculations and into real-world applications, like creating different codes or seating arrangements.