Problem 41 - College Algebra

Question

For the following exercises, evaluate the factorial. $$ \frac{12 !}{6 !} $$

Step-by-Step Solution

Answer

The value of \(\frac{12!}{6!}\) is 665280.

Step 1: Understanding Factorials

A factorial, denoted as \(!\), is the product of all positive integers up to a given number. For example, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1\).Here, we need to evaluate \(\frac{12!}{6!}\).

Step 2: Simplifying the Expression

We express both factorials and cancel the common terms. \[ \frac{12!}{6!} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6!}{6!} \] The \(6!\) in the numerator and denominator cancel each other out.

Step 3: Evaluating the Remaining Product

Now, we need to calculate the product of the remaining numbers:\[ 12 \times 11 \times 10 \times 9 \times 8 \times 7 \]Calculate it step by step or with a calculator for accuracy.

Step 4: Calculating the Final Answer

Perform the multiplication:\[ 12 \times 11 = 132 \]\[ 132 \times 10 = 1320 \]\[ 1320 \times 9 = 11880 \]\[ 11880 \times 8 = 95040 \]\[ 95040 \times 7 = 665280 \]Thus, the value of \( \frac{12!}{6!} \) is 665280.

Key Concepts

Evaluating Factorials

Factorials are essential in mathematics, particularly in permutations and combinations. When you see the factorial symbol \( ! \), it instructs us to multiply a number by all the whole numbers less than it, down to 1.
For example, \(4!\) ("four factorial") means \(4 \times 3 \times 2 \times 1 = 24\).
This concept becomes incredibly useful when you need to calculate permutations or probabilities. To evaluate a factorial, we simply continue multiplying until we reach 1.

A positive integer factorial then follows this pattern:

• \(n! = n \times (n-1) \times (n-2) \dots \times 3 \times 2 \times 1\)

Understanding this pattern is crucial when evaluating problems involving factorials, like \( \frac{12!}{6!} \), as it allows us to cancel out the common parts of the numerator and denominator, greatly simplifying our calculations.

Simplifying Expressions

Simplifying expressions with factorials can often save a lot of time and effort. When dealing with a fraction containing factorials, like \( \frac{12!}{6!} \), identifying and cancelling common terms is essential.
Here's how it works:

Start by expanding the factorials. \(12!\) is the product of numbers from 12 down to 1, while \(6!\) is the product from 6 down to 1:

• \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

Notice the sequence \(6!\) appears in both numerator and denominator. By cancelling \(6!\), only numbers unique to \(12!\) remain:

• \(\frac{12!}{6!} = 12 \times 11 \times 10 \times 9 \times 8 \times 7\)

This cancellation reduces the complexity of the problem, making it straightforward to solve.

Multiplication Steps

Once we've simplified our expression, we're left with a series of numbers to multiply. Each step of multiplication should be calculated methodically to avoid errors, especially when dealing with large numbers.
Following these steps can help:

• Start with the first two numbers: \(12 \times 11 = 132\).
  
• Progressively add one number at a time, multiplying the result: \(132 \times 10 = 1320\).
  
• Continue with: \(1320 \times 9 = 11880\), then \(11880 \times 8 = 95040\), and finally \(95040 \times 7 = 665280\).
  

This systematic approach to multiplication ensures accuracy at each step. Using a calculator can help verify the results, but it's also valuable to practice and improve your mental math skills. These steps conclude the evaluation by confirming that \(\frac{12!}{6!}\) equals 665280.