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Chapter 5: Problem 14

Find the value of each permutation. $$_7 P_{7}$$

Short Answer

Expert verified
The value of \(_7 P_{7}\) is 5040.

Step by step solution

01

Understand the Permutation Formula

Permutations are calculated using the formula }}
02

Apply the Permutation Formula

Substitute the given values into the formula: \(n = 7\) and \(r = 7\). Thus, the expression becomes $$\frac{7!}{(7-7)!}$$.
03

Simplify the Factorials

First, compute the factorials. $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040.$$ And \(0! = 1\) by definition. Plug these values back into the formula: $$\frac{7!}{0!} = 5040.$$.
04

Compute the Final Value

Perform the division: $$\frac{5040}{1} = 5040.$$.

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Key Concepts

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

Factorial
In mathematics, understanding factorials is key to grasping more complex topics like permutations. A factorial, denoted as \( n! \), is the product of all positive integers up to the number \( n \).
For example:
  • \( 3! = 3 \times 2 \times 1 = 6 \)
  • \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
  • \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

The special case \( 0! \) is defined to be 1 by convention. This is important in many mathematical formulas and ensures they work correctly for all values of \( n \). When calculating permutations, you will frequently encounter factorials, so it's essential to practice recognizing and computing them efficiently.
Permutation Formula
Permutations deal with the arrangement of objects in a specific order. The permutation formula is useful for problems where the order in which objects are arranged is important.
The general formula for the number of permutations of \( n \) objects taken \( r \) at a time is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
This formula calculates how many different ways you can arrange \( r \) objects out of a total of \( n \) objects.
To see this in action, let's use the example from the exercise with \( n = 7 \) and \( r = 7 \):
  • Substitute \( n \) and \( r \) into the formula: \[ P(7, 7) = \frac{7!}{(7-7)!} = \frac{7!}{0!} \]
  • We already know that \( 7! = 5040 \) and \( 0! = 1 \) by definition. So, \[ \frac{5040}{1} = 5040 \]

This demonstrates that there are 5040 different ways to arrange 7 objects taken 7 at a time.
Combinatorics
Combinatorics is a branch of mathematics focusing on counting, arranging, and combination of objects. Understanding combinatorics is essential for solving complex problems in probability, statistics, and computer science.
Key concepts include:
  • Permutations: The number of ways to arrange a set of objects where order matters.
  • Combinations: The number of ways to choose objects from a set where order does not matter. The formula for combinations of \( n \) objects taken \( r \) at a time is \ \frac{n!}{r!(n-r)!} \

When working with permutations and combinations, practice visualizing different ways items can be arranged or selected. This helps deepen your understanding and makes solving these problems much easier.
By mastering these fundamental concepts, you'll be well-equipped to tackle more advanced topics in mathematics and beyond.

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Most popular questions from this chapter

According to the National Center for Health Statistics, there is a \(23.4 \%\) probability that a randomly selected resident of the United States aged 25 years or older is a smoker. In addition, there is a \(21.7 \%\) probability that a randomly selected resident of the United States aged 25 years or older is female, given that he or she smokes. What is the probability that a randomly selected resident of the United States aged 25 years or older is female and smokes? Would it be unusual to randomly select a resident of the United States aged 25 years or older who is female and smokes?

Clothing Options A man has six shirts and four ties. Assuming that they all match, how many different shirt-and-tie combinations can he wear?

Suppose that \(E\) and \(F\) are two events and that \(P(E \text { and } F)=0.6\) and \(P(E)=0.8 .\) What is \(P(F | E) ?\)

Among 21 - to 25 -year-olds, \(29 \%\) say they have driven while under the influence of alcohol. Suppose three \(21-\) to 25 -year-olds are selected at random. Source: U.S. Department of Health and Human Services, reported in USA Today (a) What is the probability that all three have driven while under the influence of alcohol? (b) What is the probability that at least one has not driven while under the influence of alcohol? (c) What is the probability that none of the three has driven while under the influence of alcohol? (d) What is the probability that at least one has driven while under the influence of alcohol?

Suppose that \(E\) and \(F\) are two events and that \(P(E)=0.8\) and \(P(F | E)=0.4 .\) What is \(P(E \text { and } F) ?\)
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