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Chapter 11: Problem 103

Calculate the requested quantity. $$ 8 ! $$

Short Answer

Expert verified
The factorial of 8, denoted as \(8!\), is 40320.

Step by step solution

01

Understanding Factorial

Factorial of a number is the product of all positive integers less than or equal to that number. It is denoted by '!'. For example, the factorial of 5 is \(5*4*3*2*1\). So essentially when we say '8!', it means the product of all numbers starting from 8 and decreasing till 1.
02

Calculating the Factorial of 8

Now we implement the definition of factorial to calculate 8!. We perform the multiplication: \(8! = 8*7*6*5*4*3*2*1\).
03

Performing the multiplication

Performing the multiplication gives the result: \(8*7*6*5*4*3*2*1 = 40320\). Thus, \(8! = 40320\).

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Key Concepts

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

Understanding Multiplication
Multiplication is one of the fundamental operations in mathematics. It involves combining groups of equal size together. For instance, if you have 3 groups of apples and each group has 4 apples, multiplying 3 by 4 gives you the total number of apples, which is 12. In essence, multiplication is repeated addition. It's like adding the same number several times. Calculating a factorial, such as 8!, involves multiplying a series of descending numbers, starting from the given number down to 1.
  • The multiplication process is sequential. You multiply 8 by 7, then multiply the result by 6, and continue until 1.
  • Understanding this process can simplify calculations significantly.
Efficient multiplication is key to solving problems involving larger numbers, especially when dealing with factorials where many multiplications are required.
Exploring Number Theory
Number theory is a branch of mathematics devoted to studying the properties and relationships of numbers, particularly integers. Factorials, like 8!, are often explored within this framework. They have intriguing properties and applications in number theory and beyond.
  • Factorials grow quite rapidly. For instance, 8! is already 40,320, showing how quickly numbers can expand.
  • They are closely related to permutations and combinations, fundamental concepts in number theory used to determine possible arrangements or selections of items.
Understanding factorials helps develop a deeper grasp of how numbers relate and interact with one another, paving the way for advanced mathematical study.
Role of Factorials in Mathematics Education
Factorials play an essential role in mathematics education, particularly in developing problem-solving skills and understanding mathematical order and sequencing. They are introduced relatively early in the educational journey and serve as a building block for more complex topics.
  • Understanding the factorial notation and how to compute it prepares students for more abstract mathematical concepts like binomial expansions and calculus.
  • Factorials also aid in teaching students about efficiency and order in calculations, crucial for tackling more advanced math problems.
Using factorials, educators can illustrate mathematical principles in a concrete way, cementing core concepts and encouraging logical thought processes in students.

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Most popular questions from this chapter

During the \(2010-11\) NBA season, Ray Allen of the Boston Celtics had a free throw shooting percentage of \(0.881 .\) Assume that the probability Ray Allen makes any given free throw is fixed at 0.881 , and that free throws are independent. (a) If Ray Allen shoots two free throws, what is the probability that he makes both of them? (b) If Ray Allen shoots two free throws, what is the probability that he misses both of them? (c) If Ray Allen shoots two free throws, what is the probability that he makes exactly one of them?

Calculate the mean and standard deviation of the binomial random variable. A binomial random variable with \(n=30\) and \(p=0.5\)

Suppose that past experience shows that about \(10 \%\) of passengers who are scheduled to take a particular flight fail to show up. For this reason, airlines sometimes overbook flights, selling more tickets than they have seats, with the expectation that they will have some no shows. Suppose an airline uses a small jet with seating for 30 passengers on a regional route and assume that passengers are independent of each other in whether they show up for the flight. Suppose that the airline consistently sells 32 tickets for every one of these flights. (a) On average, how many passengers will be on each flight? (b) How often will they have enough seats for all of the passengers who show up for the flight?

State whether the process described is a discrete random variable, is a continuous random variable, or is not a random variable. Draw one M\&M from a bag. Observe whether it is blue, green, brown, orange, red, or yellow.

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.4, P(B)=0.3\), and \(P(A\) and \(B)=0.1\). Are events \(\mathrm{A}\) and \(\mathrm{B}\) disjoint?
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