Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\frac{6 !}{5 !}$$

Step-by-Step Solution

Answer

6

Step 1: Understand Factorial Notation

The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 2: Break Down the Given Expression

The given expression is \( \frac{6!}{5!} \). This means you need to calculate the factorial of 6 and the factorial of 5, and then divide the two results.

Step 3: Calculate the Factorials

Calculate \( 6! \), which is \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \), and \( 5! \), which is \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 4: Perform the Division

Substitute the factorial values into the expression: \( \frac{720}{120} \). Simplify the division to obtain the result \( 6 \).

Key Concepts

Factorial Notation

Factorial notation is a useful concept in mathematics, used to represent a product of an integer and all the positive integers below it. When you see a number with an exclamation mark next to it, like 6!, this tells you that you need to multiply all whole numbers from 6 down to 1. This is a simple, yet powerful tool, especially in permutations and combinations where arrangements are calculated.
For example:

• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24

Factorials grow very rapidly, which makes them useful in calculating probabilities and arrangements but can also become large quickly! Understanding factorial notation helps in decomposing complex mathematical problems into more manageable pieces.

Integer Operations

Integer operations without a calculator can be simplified by reducing or breaking down expressions. In the case of factorials, you often deal with large products that can be simplified through division or multiplication. Let's take the expression \( \frac{6!}{5!} \) as an example.

Instead of calculating each factorial entirely, notice that:

• 6! = 6 × 5!
• 5! = 5 × 4 × 3 × 2 × 1

By recognizing overlapping terms, for instance, the \( 5! \) in both the numerator and the denominator, we can cancel them out:

• \( \frac{6!}{5!} = \frac{6 \times 5!}{5!} = 6 \)

This simplification makes the operation quicker and avoids unnecessary calculations. Such insights are crucial for efficient mathematical problem-solving.

Calculation Without Calculator

Calculating without a calculator might seem daunting, but with factorials and integer operations, it can be straightforward using some techniques. First, break down the problem into smaller steps that can be managed mentally or by simple arithmetic.

If you face something like \( \frac{6!}{5!} \), notice the simplicity:

• Identify shared terms in fractions and cancel them out. Here, 5! appears in both parts.
• Simplify the expression to just the numbers needed for the problem, like reducing \( \frac{6!}{5!} \) to 6.

Additionally, understanding the properties of numbers and repeated terms can make problems more approachable without a calculator. This helps in reinforcing arithmetic skills and deepening your number sense, making complex math tasks much more manageable. With practice, these operations can become intuitive!