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Chapter 5: Problem 13

Find the value of each permutation. $$_{4} P_{4}$$

Short Answer

Expert verified
The value of \({}_{4} P_{4}\) is \(24\).

Step by step solution

01

Understanding the Permutation Notation

The notation \({}_{n} P_{r}\) represents the number of permutations of \(r\) objects from a set of \(n\) objects. Here, \(n = 4\) and \(r = 4\).
02

Permutation Formula

The formula for permutations is given by: \({}_{n} P_{r} = \frac{n!}{(n-r)!}\).
03

Substitute the Values

Substitute \(n = 4\) and \(r = 4\) into the formula: \({}_{4} P_{4} = \frac{4!}{(4-4)!}\).
04

Calculate the Factorials

First, compute the factorials: \(4! = 4 \times 3 \times 2 \times 1 = 24\) and \((4-4)! = 0! = 1\) (by definition, \(0!\) is \(1\)).
05

Simplify the Expression

Substitute the computed factorials back into the expression: \({}_{4} P_{4} = \frac{24}{1} = 24\).

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Key Concepts

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

Permutations
Permutations tell us how many different ways we can arrange a set of objects. Simply put, if you have a certain number of items and you want to know how many different ways you can order these items, you are calculating permutations.
For example, if you have 4 unique objects, the permutations will show how many different ways you can arrange those 4 objects in order. Each arrangement is called a permutation.
The number of permutations grows rapidly as the number of items increases. This is why understanding how to compute permutations is very useful in mathematics and various applications such as solving problems in scheduling, cryptography, and more.
Factorial Calculation
Factorial calculation is essential for finding permutations. A factorial is denoted by an exclamation mark (!), and it represents the product of all positive integers up to a given number.
For instance, for any positive integer n: \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \).
Here are some examples:
  • \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
  • \[ 3! = 3 \times 2 \times 1 = 6 \]
  • \[ 2! = 2 \times 1 = 2 \]
  • \[ 1! = 1 \]
Additionally, by mathematical definition, \( 0! \) is defined as 1.
Factorial calculations are foundational in permutations as they help determine both the numerator and denominator in permutation formulas.
Permutation Formula
The permutation formula is used to calculate the number of ways to arrange a subset of objects from a larger set. The general formula for permutations is given by: \[ {}_n P_r = \frac{n!}{(n-r)!} \] In this formula:
  • \( n \) is the total number of objects
  • \( r \) is the number of objects to be arranged
For instance, if you want to find the number of ways to arrange 4 objects out of 4, you use the formula \[ {}_4 P_4 = \frac{4!}{(4-4)!} = \frac{4!}{0!} = \frac{24}{1} = 24 \] This shows that there are 24 different ways to arrange 4 objects.
Mathematical Notation
Mathematical notation helps standardize and simplify the expression of mathematical concepts. In permutations, it is critical to understand the symbols and notation used. Here are some key notations:
  • \( {}_nP_r \): This represents the number of permutations of \( r \) objects from a set of \( n \) objects.
  • \( n! \): This denotes the factorial of \( n \), meaning the product of all positive integers up to \( n \).
Understanding notation helps in quickly decoding problems and applying the correct formulas. It also makes communication of complex mathematical ideas clearer and more concise.

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Most popular questions from this chapter

Suppose a computer chip company has just shipped 10,000 computer chips to a computer company. Unfortunately, 50 of the chips are defective. (a) Compute the probability that two randomly selected chips are defective using conditional probability. (b) There are 50 defective chips out of 10,000 shipped. The probability that the first chip randomly selected is defective is \(\frac{50}{10,000}=0.005=0.5 \% .\) Compute the probability that two randomly selected chips are defective under the assumption of independent events. Compare your results to part (a). Conclude that, when small samples are taken from large populations without replacement, the assumption of independence does not significantly affect the probability.

For the month of June in the city of Chicago, \(37 \%\) of the days are cloudy. Also in the month of June in the city of Chicago, \(21 \%\) of the days are cloudy and rainy. What is the probability that a randomly selected day in June will be rainy if it is cloudy?

The data in the following table show the results of a national study of 137,243 U.S. men that investigated the association between cigar smoking and death from cancer. Note: Current cigar smoker means "cigar smoker at time of death." $$\begin{array}{|l|c|}\hline & \text { Died from cancer } & \text { Did not die from cancer } \\ \hline \text { Never smoked cigars } & 782 & 120,747 \\\\\hline \text { Former cigar smoker } & 91 & 7,757 \\ \hline \text { Current cigar smoker } & 141 & 7,725 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected individual from the study who died from cancer was a former cigar smoker? (b) What is the probability that a randomly selected individual from the study who was a former cigar smoker died from cancer?

Find the value of each combination. $$_{12} C_{3}$$

A woman has five blouses and three skirts. Assuming that they all match, how many different outfits can she wear?
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