/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Evaluate each expression. $$\f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 16

Evaluate each expression. $$\frac{9 !}{6 !}$$

Short Answer

Expert verified
The value of \(\frac{9!}{6!}\) is 504.

Step by step solution

01

Understand the Factorial Notation

A factorial, represented by the exclamation mark (!), is the product of an integer and all the integers below it. For example, 6! (6 factorial) is the product of all integers from 1 to 6.
02

Write out the Factorials

First, write out the factorials for both the numerator and the denominator of the given fraction: \[9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\] and \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\]
03

Simplify the Fraction

Next, simplify the fraction \[ \frac{9!}{6!} \] by canceling out the common terms in the numerator and the denominator. Since 6! is common in both, it becomes: \[\frac{9 \times 8 \times 7 \times 6!}{6!} = 9 \times 8 \times 7 = 504\]
04

Perform the Multiplication

Finally, multiply the remaining numbers: \[9 \times 8 = 72\] and then \[72 \times 7 = 504\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Factorial Computation
Factorials are a fundamental concept in mathematics, especially in areas like combinatorics and probability. A factorial, represented by an exclamation mark (!), is the product of an integer and all the positive integers below it. For example, if you have 5!, this means you multiply 5 by all the numbers below it:

5! = 5 脳 4 脳 3 脳 2 脳 1 = 120

In our exercise, we need to work with 9! and 6!. Writing them down:

9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1

6! = 6 脳 5 脳 4 脳 3 脳 2 脳 1

This will help us to see the common factors and simplify our calculations.
Fraction Simplification
Next, let's simplify the fraction \(\frac{9!}{6!}\).

First, observe that 6! appears in both the numerator (top) and the denominator (bottom). This means they can be canceled out:

\(\frac{9!}{6!}\) = \(\frac{9 脳 8 脳 7 脳 6!}{6!}\).

Here, 6! in the numerator and denominator cancel each other out, leaving us with:

9 脳 8 脳 7

By reducing fractions in this way, complex calculations become much simpler. We鈥檙e left with a straightforward multiplication of three numbers.
Multiplicative Properties
The final step involves basic multiplication. Now, we just need to multiply the numbers we were left with after simplification:

Let's start with:

9 脳 8 = 72

Then, take that result and multiply it by 7:

72 脳 7 = 504

So, the value of \(\frac{9!}{6!}\) is 504. Understanding how to break down and manipulate such problems using multiplicative properties is crucial. It allows you to handle large numbers and complex expressions with ease.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

List all equally likely outcomes in the sample space for each of the following experiments. A pair of coins is tossed (4 outcomes).

The following problems may involve combinations, permutations, or the fundamental counting principle. Marching Bands In how many ways can four marching bands and four floats line up for a parade if a marching band must lead the parade and two bands cannot march next to one another?

Prove that \(\sum_{i=0}^{n}\left(\begin{array}{l}n \\\ i\end{array}\right)=2^{n}\) for any positive integer \(n\)

Write the complete binomial expansion for each of the following powers of a binomial. $$(x+5)^{2}$$

Explain why \(C(n, r)=C(n, n-r)\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.