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Chapter 5: Problem 8

Find the value of each factorial. \(12 !\)

Short Answer

Expert verified
The value of 12! is 479001600.

Step by step solution

01

Understand the Concept of a Factorial

A factorial of a number, denoted as n!, is the product of all positive integers less than or equal to n. For example, 5! is equal to 5 脳 4 脳 3 脳 2 脳 1.
02

Set Up 12!

Write out the factorial expression for n = 12. So, 12! is written as 12 脳 11 脳 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
03

Calculate the Product

Multiply these numbers step-by-step: 12 脳 11 = 132, 132 脳 10 = 1320, 1320 脳 9 = 11880, 11880 脳 8 = 95040, 95040 脳 7 = 665280, 665280 脳 6 = 3991680, 3991680 脳 5 = 19958400, 19958400 脳 4 = 79833600, 79833600 脳 3 = 239500800, 239500800 脳 2 = 479001600, and finally, 479001600 脳 1 = 479001600.

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Key Concepts

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

Factorial Definition
A factorial is a mathematical operation where you take a positive integer n and multiply it by every positive integer less than it. It is represented by the symbol n!. For example, if n is 5, you write it as 5!. This means you multiply 5 by 4, then 4 by 3, and so on, until you reach 1. So, 5! = 5 脳 4 脳 3 脳 2 脳 1. This pattern holds true for any positive integer n.
Factorials grow very quickly in size. For instance, while 5! is just 120, something like 10! is already 3,628,800. This rapid growth makes factorials useful in areas like combinatorics and probability.
Mathematical Operations
When calculating a factorial, you are performing a series of multiplications. This involves consecutive steps where you multiply descending integers. To make this more clear, let鈥檚 break down a smaller factorial, such as 4!.

For example:
4! = 4 脳 3 脳 2 脳 1

  • First, you multiply 4 by 3 to get 12.
  • Then, multiply that result by 2 to get 24.
  • Finally, multiply that result by 1 to still get 24.

Notice each step builds on the previous step. This method helps in making sure you don鈥檛 skip a number or make an error in your calculations.
Multiplication Steps
To calculate a larger factorial, such as 12!, follow these multiplication steps sequentially:

1. Start with the largest number: 12
2. Multiply it by the next smallest number: 12 脳 11 = 132
3. Continue this pattern: 132 脳 10 = 1320
4. Each step involves multiplying the current result by the next smallest number: 1320 脳 9 = 11880

To finalize:
  • 11880 脳 8 = 95040
  • 95040 脳 7 = 665280
  • 665280 脳 6 = 3991680
  • 3991680 脳 5 = 19958400
  • 19958400 脳 4 = 79833600
  • 79833600 脳 3 = 239500800
  • 239500800 脳 2 = 479001600
  • 479001600 脳 1 = 479001600


Following these steps carefully ensures you reach the correct result. This sequence also illustrates the rapid growth of factorial calculations, even with a reasonably small starting number like 12.

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Most popular questions from this chapter

For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than \(0.9999 ?\)

According to the U.S. National Center for Health Statistics, \(0.15 \%\) of deaths in the United States are 25 - to 34-year-olds whose cause of death is cancer. In addition, \(1.71 \%\) of all those who die are \(25-34\) years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been \(25-34\) years old?

In finance, a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage has a probability of 0.01 of default. (a) What is the probability a randomly selected mortgage will not default (that is, pay off)? (b) What is the probability a bundle of five randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment in which all five mortgages do not default. (c) What is the probability the derivative becomes worthless? That is, at least one of the mortgages defaults? (d) In part (b), we made the assumption that the likelihood of default is independent. Do you believe this is a reasonable assumption? Explain.

Ken and Dorothy like to fly to Colorado for ski vacations. Sometimes, however, they are late for their flight. On the air carrier they prefer to fly, the probability that luggage gets lost is 0.012 for luggage checked at least one hour prior to departure. However, the probability luggage gets lost is 0.043 for luggage checked within one hour of departure. Are the events "luggage check time" and "lost luggage" independent? Explain.

(See Example 10.) How many distinguishable DNA sequences can be formed using three As, two Cs, two Gs, and three Ts?
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