Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the factorial expression. $$\frac{5 !}{7 !}$$

Step-by-Step Solution

Answer

The simplified form of the given factorial fraction \( \frac{5 !}{7 !} \) is \( \frac{1}{42} \).

Step 1: Understanding Factorials

Understand the factorial operation. The factorial of a number \( n \) denoted as \( n! \), is a product of all positive integers less than or equal to \( n \). For instance, 5! equals 5 * 4 * 3 * 2 * 1.

Step 2: Replace Larger Factorial

Replace the larger factorial 7! using factorial property. Express 7! in terms of 5!. Remember that 7! = 7 * 6 * 5!, so the expression becomes: \( \frac{5 !}{7*6*5 !} \)

Step 3: Simplify Expression

Now, we can cancel out 5!, which appears in both the top and bottom of our fraction. After simplifying, the expression becomes: \( \frac{1}{7*6} \)

Key Concepts

Factorial Operation

Factorials are a fundamental concept in mathematics, often symbolized with an exclamation mark. The factorial of a number, denoted as \( n! \), is computed by multiplying the number \( n \) by every positive number less than itself. This series of multiplications will always end at 1.

• Example: The factorial of 5, written as \( 5! \), is calculated as 5 times 4 times 3 times 2 times 1, resulting in 120.
• The special case is \( 0! \), which is equal to 1, by definition.

Factorials grow extremely fast. As a number gets larger, the factorial of that number increases exponentially. Understanding how to compute and manipulate these products of an integer and all its predecessors is key to simplifying factorial expressions efficiently.

Factorial Property

An interesting and very useful property of factorials is the relationship each factorial number has with its predecessor. This property states that any factorial \( n! \) can be expressed as \( n \times (n-1)! \). This means each factorial number contains the factorial of the number just before it.
This property proves especially helpful when simplifying fractions involving factorials. For example:

• \( 7! = 7 \times 6 \times 5! \)
• This ability to "factor out" factorials is what allows us to simplify larger factorials in terms of smaller ones.

This property allows us to express a larger factorial in terms of a smaller one and cancel out common terms in the numerator and the denominator. Recognizing and using this recursive relationship is crucial for breaking down and simplifying factorial-based expressions.

Simplifying Fractions

When dealing with fractions like \( \frac{5!}{7!} \), you can simplify them by canceling out common terms in the numerator and the denominator. The factorial property allows you to rewrite the expression, helping in simplification.
Here’s how you can simplify such expressions:

• First, express the larger factorial in terms of the smaller one. For instance, in our expression, \( 7! \) is rewritten as \( 7 \times 6 \times 5! \).
• Next, cancel out the common factorial term. Here, \( 5! \) on both the numerator and denominator cancels out.
• This results in \( \frac{1}{7 \times 6} \) after simplification, which equals \( \frac{1}{42} \).

Simplifying fractions with factorials becomes straightforward when you break down the factorals until all possible cancelations have been made. This systematic approach ensures that you arrive at the simplest form of the expression efficiently and accurately.