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Chapter 9: Problem 70

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

Short Answer

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

Step by step solution

01

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.
02

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: \( \frac{5 !}{7*6*5 !} \)
03

Simplify Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

Write the first five terms of the sequence defined recursively. $$a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}$$

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{1}{n !}$$

Use the Binomial Theorem to expand and simplify the expression. \((x+y)^{5}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{n^{3}}{(n+2) !}$$

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=4, r=2$$
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