/*! 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 10 Find the value of each factorial... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

Find the value of each factorial. \(1!\)

Short Answer

Expert verified
1! = 1

Step by step solution

01

Understand the Concept of Factorial

A factorial of a number is the product of all positive integers from 1 to that number. It is denoted by the symbol '!'. For example, the factorial of 1, written as 1!.
02

Apply the Definition of Factorial

Since 1! represents the product of all positive integers up to 1, it includes only one number: 1. Therefore, 1! is simply 1.

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.

basic mathematics
Factorials are one of the fundamental concepts in basic mathematics. They are used to calculate the product of all positive integers up to a specific number. Factorials are denoted by the exclamation mark, like this: (1!).
For example:
  • The factorial of 1 is 1! = 1.
  • The factorial of 2 is 2! = 1 脳 2 = 2.
  • The factorial of 3 is 3! = 1 脳 2 脳 3 = 6.
The definition is simple, yet powerful. Factorials grow very fast. For instance, 5! = 120 and 7! = 5040.
Understanding factorials can help you grasp more complex topics in combinatorics and probability.
introductory statistics
In introductory statistics, factorials are often used when dealing with permutations and combinations. These concepts help in understanding how data can be arranged and categorized.
For example:
  • Permutations: The number of ways to arrange 'n' things in order is given by 'n!'.
  • Combinations: The number of ways to choose 'r' things from 'n' things without worrying about the order is given by \(\binom{n}{r}\), and it involves factorials.
Therefore, having a good grasp of factorials is essential for solving many problems in probability and statistics.
Simple example: If you want to know how many ways you can arrange 3 books on a shelf, you calculate 3! = 1 脳 2 脳 3 = 6 ways.
combinatorics
Combinatorics is a branch of mathematics that deals with counting, arranging, and combinations of objects. Factorials play a crucial role here.
In combinatorics:
  • Factorials are used to calculate permutations, where order matters. For example, the number of ways to arrange 4 items is 4!, which is 24.
  • Factorials are also used in combinations, where order does not matter. The formula for combinations involves dividing a factorial by another factorial.
For example, the number of ways to choose 2 items from 4 (without caring about the order) is given by \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{24}{2 \times 2} = 6. \]
Thus, understanding factorials is key to mastering topics in combinatorics.

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

How many different simple random samples of size 5 can be obtained from a population whose size is \(50 ?\)

A golf-course architect has four linden trees, five white birch trees, and two bald cypress trees to plant in a row along a fairway. In how many ways can the landscaper plant the trees in a row, assuming that the trees are evenly spaced?

In the U.S. Senate, there are 21 members on the Committee on Banking, Housing, and Urban Affairs. Nine of these 21 members are selected to be on the Subcommittee on Economic Policy. How many different committee structures are possible for this subcommittee?

In Problem \(35,\) we learned that for some diseases, such as sickle-cell anemia, an individual will get the disease only if he receives both recessive alleles. This is not always the case. For example, Huntington's disease only requires one dominant gene for an individual to contract the disease. Suppose a husband and wife, who both have a dominant Huntington's disease allele \((S)\) and a normal recessive allele \((s),\) decide to have a child. (a) List the possible genotypes of their offspring. (b) What is the probability that the offspring will not have Huntington's disease? In other words, what is the probability the offspring will have genotype \(s s ?\) Interpret this probability. (c) What is the probability that the offspring will have Huntington's disease?

Suppose a compact disk (CD) you just purchased has 13 tracks. After listening to the CD, you decide that you like 5 of the songs. The random feature on your CD player will play each of the 13 songs once in a random order. Find the probability that among the first 4 songs played (a) you like 2 of them; (b) you like 3 of them; (c) you like all 4 of them.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.