Problem 8 - Algebra for College Students

Question

Evaluate each expression. $$\frac{6 !}{5 ! 1 !}$$

Step-by-Step Solution

Answer

6

Step 1: Evaluate Factorials

Calculate the factorials involved in the expression. For \(6!\), \(5!\), and \(1!\): \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\] \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\] \[1! = 1\]

Step 2: Substitute Factorials into the Expression

Substitute the evaluated factorials into the original expression: \[\frac{6!}{5! 1!} = \frac{720}{120 \times 1}\]

Step 3: Simplify the Expression

Simplify the fraction by dividing \(720\) by \(120\): \[\frac{720}{120} = 6\]

Key Concepts

Evaluate Factorials

Understanding factorials is the first step in evaluating and simplifying expressions like \(\frac{6!}{5! 1!}\). A factorial, denoted by \(n!\), represents the product of all positive integers up to \(n\). For example:

\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]
Similarly:

\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]
And for \(1!\), it's simply:

\[1! = 1\]
Factorial notation helps in quickly understanding how numbers multiply in a sequence. This is crucial for solving combinations, permutations, and many algebraic expressions.

Simplify Fractions

Once we have the values of the factorials, the next step is to simplify the expression by substituting these values back into the original fraction. The key here is to substitute each evaluated factorial properly:

\[\frac{6!}{5! 1!} = \frac{720}{120 \times 1}\]
Here:

• \(6! = 720\)
• \(5! = 120\)
• \(1! = 1\)

This enables us to simplify the fraction more easily. Now, divide 720 by 120 to simplify:

\[\frac{720}{120} = 6\]
Always remember to check if the numbers can be further divided to simplify the fraction to its lowest form.

Step by Step Solution

Breaking down each step makes solving factorial expressions much simpler. Here is a detailed step-by-step approach to evaluate \(\frac{6!}{5! 1!}\):


Step 1: Evaluate Factorials

Calculate the factorial of each number involved:

• \(6! = 720\)
• \(5! = 120\)
• \(1! = 1\)

Step 2: Substitute Factorials into the Expression

Replace the factorials in the original fraction:

\[\frac{6!}{5! 1!} = \frac{720}{120 \times 1}\]

Step 3: Simplify the Expression

Now, simplify the fraction by dividing:

\[\frac{720}{120} = 6\]
These steps help you easily follow the logic and method to solve similar problems consistently and accurately.