Problem 4 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$\frac{6 !}{5 !}$$

Step-by-Step Solution

Answer

The value of \( \frac{6!}{5!} \) is 6.

Step 1: Understanding Factorials

The expression involves factorials. The factorial of a number, denoted as \( n! \), means multiplying all integers from 1 to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 2: Write Out the Factorials

Express both \( 6! \) and \( 5! \) in their expanded form: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) and \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 3: Simplify the Division

The expression is \( \frac{6!}{5!} \). Substitute the factorial expressions: \( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \).

Step 4: Cancel Common Terms

In the fraction, \( 5 \times 4 \times 3 \times 2 \times 1 \) appears in both the numerator and the denominator, so they cancel out. What remains is \( 6 \).

Step 5: Write the Final Solution

After cancellation, the value of the expression is \( 6 \). Thus, \( \frac{6!}{5!} = 6 \).

Key Concepts

Algebra and Its Connection to Factorials

Algebra is a fundamental branch of mathematics revolving around symbols and rules for manipulating those symbols. It is essential in solving equations and understanding mathematical relationships. Factorials, denoted by an exclamation point, like \( n! \), play a crucial role in algebra as they help in simplifying complex expressions and solving permutations and combinations problems. Factorials refer to the product of all positive integers up to a specified number \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \). This notation is an example of how algebra uses symbols to condense large multiplication sequences into simpler expressions. Understanding how to manipulate these symbols is a key skill that will help you tackle various algebraic problems.
Algae involves not just solving for unknowns but also simplifying expressions like factorials to make calculations more manageable.

Simplification in Mathematical Expressions

Simplification is essential when working with algebraic expressions, as it helps make them easier to understand and work with. When you encounter expressions involving factorials, simplification often involves reducing the expression by canceling out common terms in numerators and denominators.
Let's revisit the expression \( \frac{6!}{5!} \). By expanding both factorials:

• \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

When you break them down, you can see that both numerators and denominators share common terms. The simplification process allows you to cancel these shared parts, leaving you with a much simpler expression, \( 6 \).
This process not only reduces the computational effort but also enhances understanding of the problem structure.

Mathematical Operations with Factorials

Mathematical operations involving factorials typically include multiplication, division, and simplification, which can initially seem challenging.
In our exercise, \( \frac{6!}{5!} \), we started by performing a division of two factorial expressions. The immediate step is to recognize the structure of each factorial by writing them out. The next crucial operation is simplifying via division by canceling identical factors found in both components of the division.
By performing the division \( \frac{6!}{5!} \), students often recognize that the result is not about multiplying or calculating the full factorials but spotting how factorial division reduces due to overlapping factors.
Understanding these operations shows the elegance of mathematical simplification, making seemingly complex problems turn into manageable tasks.